https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Math+piggy&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T20:41:59ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=1981_AHSME_Problems/Problem_24&diff=1574601981 AHSME Problems/Problem 242021-07-06T19:24:01Z<p>Math piggy: /* Solution */</p>
<hr />
<div>==Problem ==<br />
If <math> \theta</math> is a constant such that <math> 0 < \theta < \pi</math> and <math> x + \dfrac{1}{x} = 2\cos{\theta}</math>, then for each positive integer <math> n</math>, <math> x^n + \dfrac{1}{x^n}</math> equals<br />
<br />
<math> \textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\theta</math><br />
<br />
== Solution ==<br />
Multiply both sides by <math>x</math> and rearrange to <math>x^2-2x\cos(\theta)+1=0</math>. Using the quadratic equation, we can solve for <math>x</math>. After some simplifying:<br />
<br />
<cmath>x=\cos(\theta) + \sqrt{\cos^2(\theta)-1}</cmath><br />
<cmath>x=\cos(\theta) + \sqrt{(-1)(\sin^2(\theta))}</cmath><br />
<cmath>x=\cos(\theta) + i\sin(\theta)</cmath><br />
<br />
Substituting this expression in to the desired <math> x^n + \dfrac{1}{x^n}</math> gives:<br />
<br />
<cmath>(\cos(\theta) + i\sin(\theta))^n + (\cos(\theta) + i\sin(\theta))^{-n}</cmath><br />
<br />
Using DeMoivre's Theorem:<br />
<br />
<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(-n\theta) + i\sin(-n\theta)</cmath><br />
<br />
Because <math>\cos</math> is even and <math>\sin</math> is odd:<br />
<br />
<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta)</cmath><br />
<br />
<math>=\boxed{\textbf{2\cos(n\theta)}},</math><br />
<br />
which gives the answer <math>\boxed{\textbf{D}}.</math></div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=1981_AHSME_Problems/Problem_24&diff=1574591981 AHSME Problems/Problem 242021-07-06T19:22:00Z<p>Math piggy: /* Solution */</p>
<hr />
<div>==Problem ==<br />
If <math> \theta</math> is a constant such that <math> 0 < \theta < \pi</math> and <math> x + \dfrac{1}{x} = 2\cos{\theta}</math>, then for each positive integer <math> n</math>, <math> x^n + \dfrac{1}{x^n}</math> equals<br />
<br />
<math> \textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\theta</math><br />
<br />
== Solution ==<br />
Multiply both sides by <math>x</math> and rearrange to <math>x^2-2x\cos(\theta)+1=0</math>. Using the quadratic equation, we can solve for <math>x</math>. After some simplifying:<br />
<br />
<cmath>x=\cos(\theta) + \sqrt{\cos^2(\theta)-1}</cmath><br />
<cmath>x=\cos(\theta) + \sqrt{(-1)(\sin^2(\theta))}</cmath><br />
<cmath>x=\cos(\theta) + i\sin(\theta)</cmath><br />
<br />
Substituting this expression in to the desired <math> x^n + \dfrac{1}{x^n}</math> gives:<br />
<br />
<cmath>(\cos(\theta) + i\sin(\theta))^n + (\cos(\theta) + i\sin(\theta))^{-n}</cmath><br />
<br />
Using DeMoivre's Theorem:<br />
<br />
<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(-n\theta) + i\sin(-n\theta)</cmath><br />
<br />
Because <math>\cos</math> is even and <math>\sin</math> is odd:<br />
<br />
<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta)</cmath><br />
<cmath>=\boxed{\textbf{2\cos(n\theta)}},</cmath><br />
<br />
which gives the answer <math>\boxed{\textbf{D}}.</math></div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=1981_AHSME_Problems/Problem_24&diff=1574571981 AHSME Problems/Problem 242021-07-06T19:20:47Z<p>Math piggy: /* Solution */</p>
<hr />
<div>==Problem ==<br />
If <math> \theta</math> is a constant such that <math> 0 < \theta < \pi</math> and <math> x + \dfrac{1}{x} = 2\cos{\theta}</math>, then for each positive integer <math> n</math>, <math> x^n + \dfrac{1}{x^n}</math> equals<br />
<br />
<math> \textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\theta</math><br />
<br />
== Solution ==<br />
Multiply both sides by <math>x</math> and rearrange to <math>x^2-2x\cos(\theta)+1=0</math>. Using the quadratic equation, we can solve for <math>x</math>. After some simplifying:<br />
<br />
<cmath>x=\cos(\theta) + \sqrt{\cos^2(\theta)-1}</cmath><br />
<cmath>x=\cos(\theta) + \sqrt{(-1)(\sin^2(\theta))}</cmath><br />
<cmath>x=\cos(\theta) + i\sin(\theta)</cmath><br />
<br />
Substituting this expression in to the desired <math> x^n + \dfrac{1}{x^n}</math> gives:<br />
<br />
<cmath>(\cos(\theta) + i\sin(\theta))^n + (\cos(\theta) + i\sin(\theta))^{-n}</cmath><br />
<br />
Using DeMoivre's Theorem:<br />
<br />
<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(-n\theta) + i\sin(-n\theta)</cmath><br />
<br />
Because <math>\cos</math> is even and <math>\sin</math> is odd:<br />
\begin{align*}<br />
&=\cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta) \\<br />
&=\boxed{\textbf{2\cos(n\theta)}},<br />
\end{align*}<br />
<br />
which gives the answer <math>\boxed{\textbf{D}}.</math></div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=1981_AHSME_Problems/Problem_24&diff=1574561981 AHSME Problems/Problem 242021-07-06T19:18:08Z<p>Math piggy: /* Solution */</p>
<hr />
<div>==Problem ==<br />
If <math> \theta</math> is a constant such that <math> 0 < \theta < \pi</math> and <math> x + \dfrac{1}{x} = 2\cos{\theta}</math>, then for each positive integer <math> n</math>, <math> x^n + \dfrac{1}{x^n}</math> equals<br />
<br />
<math> \textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\theta</math><br />
<br />
== Solution ==<br />
Multiply both sides by <math>x</math> and rearrange to <math>x^2-2x\cos(\theta)+1=0</math>. Using the quadratic equation, we can solve for <math>x</math>. After some simplifying:<br />
<br />
<cmath>x=\cos(\theta) + \sqrt{\cos^2(\theta)-1}</cmath><br />
<cmath>x=\cos(\theta) + \sqrt{(-1)(\sin^2(\theta))}</cmath><br />
<cmath>x=\cos(\theta) + i\sin(\theta)</cmath><br />
<br />
Substituting this expression in to the desired <math> x^n + \dfrac{1}{x^n}</math> gives:<br />
<br />
<cmath>(\cos(\theta) + i\sin(\theta))^n + (\cos(\theta) + i\sin(\theta))^{-n}</cmath><br />
<br />
Using DeMoivre's Theorem:<br />
<br />
<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(-n\theta) + i\sin(-n\theta)</cmath><br />
<br />
Because <math>\cos</math> is even and <math>\sin</math> is odd:<br />
<br />
<cmath>=\cos(n\theta) + i\sin(n\theta) + \cos(n\theta) - i\sin(n\theta)</cmath><br />
<cmath>=\boxed{\textbf{2\cos(n\theta)}},</cmath><br />
<br />
which gives the answer <math>\boxed{\textbf{D}}.</math></div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=1983_AIME_Problems/Problem_11&diff=1284531983 AIME Problems/Problem 112020-07-17T14:30:34Z<p>Math piggy: /* Solution 4 */</p>
<hr />
<div>== Problem ==<br />
The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to the base and has length <math>2s</math>. All other edges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid?<br />
<center><asy><br />
size(180);<br />
import three; pathpen = black+linewidth(0.65); pointpen = black;<br />
currentprojection = perspective(30,-20,10);<br />
real s = 6 * 2^.5;<br />
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);<br />
draw(A--B--C--D--A--E--D);<br />
draw(B--F--C);<br />
draw(E--F);<br />
label("A",A,W);<br />
label("B",B,S);<br />
label("C",C,SE);<br />
label("D",D,NE);<br />
label("E",E,N);<br />
label("F",F,N);<br />
</asy></center> <!-- Asymptote replacement for Image:1983Number11.JPG by bpms --><br />
<br />
== Solutions ==<br />
<br />
=== Solution 1 ===<br />
First, we find the height of the solid by dropping a perpendicular from the midpoint of <math>AD</math> to <math>EF</math>. The hypotenuse of the triangle formed is the [[median]] of equilateral triangle <math>ADE</math>, and one of the legs is <math>3\sqrt{2}</math>. We apply the Pythagorean Theorem to deduce that the height is <math>6</math>.<br />
<center><asy><br />
size(180);<br />
import three; pathpen = black+linewidth(0.65); pointpen = black; pen d = linewidth(0.65); pen l = linewidth(0.5);<br />
currentprojection = perspective(30,-20,10);<br />
real s = 6 * 2^.5;<br />
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);<br />
triple Aa=(E.x,0,0),Ba=(F.x,0,0),Ca=(F.x,s,0),Da=(E.x,s,0);<br />
draw(A--B--C--D--A--E--D);<br />
draw(B--F--C);<br />
draw(E--F); <br />
draw(B--Ba--Ca--C,dashed+d);<br />
draw(A--Aa--Da--D,dashed+d);<br />
draw(E--(E.x,E.y,0),dashed+l);<br />
draw(F--(F.x,F.y,0),dashed+l);<br />
draw(Aa--E--Da,dashed+d);<br />
draw(Ba--F--Ca,dashed+d);<br />
label("A",A,S);<br />
label("B",B,S);<br />
label("C",C,S);<br />
label("D",D,NE);<br />
label("E",E,N);<br />
label("F",F,N);<br />
label("$12\sqrt{2}$",(E+F)/2,N);<br />
label("$6\sqrt{2}$",(A+B)/2,S);<br />
label("6",(3*s/2,s/2,3),ENE);<br />
</asy></center><br />
Next, we complete the figure into a triangular prism, and find its volume, which is <math>\frac{6\sqrt{2}\cdot 12\sqrt{2}\cdot 6}{2}=432</math>.<br />
<br />
Now, we subtract off the two extra [[pyramid]]s that we included, whose combined volume is <math>2\cdot \left( \frac{6\sqrt{2}\cdot 3\sqrt{2} \cdot 6}{3} \right)=144</math>.<br />
<br />
Thus, our answer is <math>432-144=\boxed{288}</math>.<br />
<br />
=== Solution 2 ===<br />
<center><asy><br />
size(180);<br />
import three; pathpen = black+linewidth(0.65); pointpen = black;<br />
currentprojection = perspective(30,-20,10);<br />
real s = 6 * 2^.5;<br />
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6),G=(s/2,-s/2,-6),H=(s/2,3*s/2,-6);<br />
draw(A--B--C--D--A--E--D);<br />
draw(B--F--C);<br />
draw(E--F);<br />
draw(A--G--B,dashed);draw(G--H,dashed);draw(C--H--D,dashed);<br />
label("A",A,(-1,-1,0));<br />
label("B",B,( 2,-1,0));<br />
label("C",C,( 1, 1,0));<br />
label("D",D,(-1, 1,0));<br />
label("E",E,(0,0,1));<br />
label("F",F,(0,0,1));<br />
label("G",G,(0,0,-1));<br />
label("H",H,(0,0,-1));<br />
</asy></center><br />
Extend <math>EA</math> and <math>FB</math> to meet at <math>G</math>, and <math>ED</math> and <math>FC</math> to meet at <math>H</math>. Now, we have a regular tetrahedron <math>EFGH</math>, which by symmetry has twice the volume of our original solid. This tetrahedron has side length <math>2s = 12\sqrt{2}</math>. Using the formula for the volume of a regular tetrahedron, which is <math>V = \frac{\sqrt{2}S^3}{12}</math>, where S is the side length of the tetrahedron, the volume of our original solid is:<br />
<br />
<math>V = \frac{1}{2} \cdot \frac{\sqrt{2} \cdot (12\sqrt{2})^3}{12} = \boxed{288}</math>.<br />
<br />
=== Solution 3 ===<br />
We can also find the volume by considering horizontal cross-sections of the solid and using calculus. As in Solution 1, we can find that the height of the solid is <math>6</math>; thus, we will integrate with respect to height from <math>0</math> to <math>6</math>, noting that each cross section of height <math>dh</math> is a rectangle. The volume is then <math>\int_0^h(wl) \ \text{d}h</math>, where <math>w</math> is the width of the rectangle and <math>l</math> is the length. We can express <math>w</math> in terms of <math>h</math> as <math>w=6\sqrt{2}-\sqrt{2}h</math> since it decreases linearly with respect to <math>h</math>, and <math>l=6\sqrt{2}+\sqrt{2}h</math> since it similarly increases linearly with respect to <math>h</math>. Now we solve:<cmath>\int_0^6(6\sqrt{2}-\sqrt{2}h)(6\sqrt{2}+\sqrt{2}h)\ \text{d}h =\int_0^6(72-2h^2)\ \text{d}h=72(6)-2\left(\frac{1}{3}\right)\left(6^3\right)=\boxed{288}</cmath>.<br />
<br />
==Solution 4==<br />
Draw an altitude from a vertex of the square base to the top edge. By using <math>30,60, 90</math> triangle ratios, we obtain that the altitude has a length of <math>3 \sqrt{6}</math>, and that little portion that hangs out has a length of <math>3\sqrt2</math>. This is a triangular pyramid with a base of <math>3\sqrt6, 3\sqrt6, 3\sqrt2</math>, and a height of <math>3\sqrt{2}</math>. Since there are two of these, we can compute the sum of the volumes of these two to be <math>72</math>. Now we are left with a triangular prism with a base of dimensions <math>3\sqrt6, 3\sqrt6, 3\sqrt2</math> and a height of <math>6\sqrt2</math>. We can compute the volume of this to be 216, and thus our answer is <math>\boxed{288}</math>.<br />
<br />
pi_is_3.141<br />
<br />
== See Also ==<br />
{{AIME box|year=1983|num-b=10|num-a=12}}<br />
<br />
[[Category:Intermediate Geometry Problems]]</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=1983_AIME_Problems/Problem_11&diff=1284521983 AIME Problems/Problem 112020-07-17T14:29:55Z<p>Math piggy: /* Solution 4 */</p>
<hr />
<div>== Problem ==<br />
The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to the base and has length <math>2s</math>. All other edges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid?<br />
<center><asy><br />
size(180);<br />
import three; pathpen = black+linewidth(0.65); pointpen = black;<br />
currentprojection = perspective(30,-20,10);<br />
real s = 6 * 2^.5;<br />
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);<br />
draw(A--B--C--D--A--E--D);<br />
draw(B--F--C);<br />
draw(E--F);<br />
label("A",A,W);<br />
label("B",B,S);<br />
label("C",C,SE);<br />
label("D",D,NE);<br />
label("E",E,N);<br />
label("F",F,N);<br />
</asy></center> <!-- Asymptote replacement for Image:1983Number11.JPG by bpms --><br />
<br />
== Solutions ==<br />
<br />
=== Solution 1 ===<br />
First, we find the height of the solid by dropping a perpendicular from the midpoint of <math>AD</math> to <math>EF</math>. The hypotenuse of the triangle formed is the [[median]] of equilateral triangle <math>ADE</math>, and one of the legs is <math>3\sqrt{2}</math>. We apply the Pythagorean Theorem to deduce that the height is <math>6</math>.<br />
<center><asy><br />
size(180);<br />
import three; pathpen = black+linewidth(0.65); pointpen = black; pen d = linewidth(0.65); pen l = linewidth(0.5);<br />
currentprojection = perspective(30,-20,10);<br />
real s = 6 * 2^.5;<br />
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);<br />
triple Aa=(E.x,0,0),Ba=(F.x,0,0),Ca=(F.x,s,0),Da=(E.x,s,0);<br />
draw(A--B--C--D--A--E--D);<br />
draw(B--F--C);<br />
draw(E--F); <br />
draw(B--Ba--Ca--C,dashed+d);<br />
draw(A--Aa--Da--D,dashed+d);<br />
draw(E--(E.x,E.y,0),dashed+l);<br />
draw(F--(F.x,F.y,0),dashed+l);<br />
draw(Aa--E--Da,dashed+d);<br />
draw(Ba--F--Ca,dashed+d);<br />
label("A",A,S);<br />
label("B",B,S);<br />
label("C",C,S);<br />
label("D",D,NE);<br />
label("E",E,N);<br />
label("F",F,N);<br />
label("$12\sqrt{2}$",(E+F)/2,N);<br />
label("$6\sqrt{2}$",(A+B)/2,S);<br />
label("6",(3*s/2,s/2,3),ENE);<br />
</asy></center><br />
Next, we complete the figure into a triangular prism, and find its volume, which is <math>\frac{6\sqrt{2}\cdot 12\sqrt{2}\cdot 6}{2}=432</math>.<br />
<br />
Now, we subtract off the two extra [[pyramid]]s that we included, whose combined volume is <math>2\cdot \left( \frac{6\sqrt{2}\cdot 3\sqrt{2} \cdot 6}{3} \right)=144</math>.<br />
<br />
Thus, our answer is <math>432-144=\boxed{288}</math>.<br />
<br />
=== Solution 2 ===<br />
<center><asy><br />
size(180);<br />
import three; pathpen = black+linewidth(0.65); pointpen = black;<br />
currentprojection = perspective(30,-20,10);<br />
real s = 6 * 2^.5;<br />
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6),G=(s/2,-s/2,-6),H=(s/2,3*s/2,-6);<br />
draw(A--B--C--D--A--E--D);<br />
draw(B--F--C);<br />
draw(E--F);<br />
draw(A--G--B,dashed);draw(G--H,dashed);draw(C--H--D,dashed);<br />
label("A",A,(-1,-1,0));<br />
label("B",B,( 2,-1,0));<br />
label("C",C,( 1, 1,0));<br />
label("D",D,(-1, 1,0));<br />
label("E",E,(0,0,1));<br />
label("F",F,(0,0,1));<br />
label("G",G,(0,0,-1));<br />
label("H",H,(0,0,-1));<br />
</asy></center><br />
Extend <math>EA</math> and <math>FB</math> to meet at <math>G</math>, and <math>ED</math> and <math>FC</math> to meet at <math>H</math>. Now, we have a regular tetrahedron <math>EFGH</math>, which by symmetry has twice the volume of our original solid. This tetrahedron has side length <math>2s = 12\sqrt{2}</math>. Using the formula for the volume of a regular tetrahedron, which is <math>V = \frac{\sqrt{2}S^3}{12}</math>, where S is the side length of the tetrahedron, the volume of our original solid is:<br />
<br />
<math>V = \frac{1}{2} \cdot \frac{\sqrt{2} \cdot (12\sqrt{2})^3}{12} = \boxed{288}</math>.<br />
<br />
=== Solution 3 ===<br />
We can also find the volume by considering horizontal cross-sections of the solid and using calculus. As in Solution 1, we can find that the height of the solid is <math>6</math>; thus, we will integrate with respect to height from <math>0</math> to <math>6</math>, noting that each cross section of height <math>dh</math> is a rectangle. The volume is then <math>\int_0^h(wl) \ \text{d}h</math>, where <math>w</math> is the width of the rectangle and <math>l</math> is the length. We can express <math>w</math> in terms of <math>h</math> as <math>w=6\sqrt{2}-\sqrt{2}h</math> since it decreases linearly with respect to <math>h</math>, and <math>l=6\sqrt{2}+\sqrt{2}h</math> since it similarly increases linearly with respect to <math>h</math>. Now we solve:<cmath>\int_0^6(6\sqrt{2}-\sqrt{2}h)(6\sqrt{2}+\sqrt{2}h)\ \text{d}h =\int_0^6(72-2h^2)\ \text{d}h=72(6)-2\left(\frac{1}{3}\right)\left(6^3\right)=\boxed{288}</cmath>.<br />
<br />
==Solution 4==<br />
Draw an altitude from a vertex of the square base to the top edge. By using <math>30,60, 90</math> triangle ratios, we obtain that the altitude has a length of <math>3 \sqrt{6}</math>, and that little portion that hangs out has a length of <math>3\sqrt2</math>. This is a triangular pyramid with a base of <math>3\sqrt6, 3\sqrt6, 3\sqrt2</math>, and a height of <math>3\sqrt{2}</math>. Since there are two of these, we can compute the sum of the volumes of these two to be <math>\72</math>. Now we are left with a triangular prism with a base of dimensions <math>3\sqrt6, 3\sqrt6, 3\sqrt2</math> and a height of <math>6\sqrt2</math>. We can compute the volume of this to be 216, and thus our answer is <math>\boxed{288}</math>.<br />
<br />
pi_is_3.141<br />
<br />
== See Also ==<br />
{{AIME box|year=1983|num-b=10|num-a=12}}<br />
<br />
[[Category:Intermediate Geometry Problems]]</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870082007 iTest Problems2017-08-12T15:50:17Z<p>Math piggy: /* Problem 11 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^2\cdot \cdot \cdot^2}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108</math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4</math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad</math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad</math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad</math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad</math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad</math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad</math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870072007 iTest Problems2017-08-12T15:50:04Z<p>Math piggy: /* Problem 11 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^2^\cdot \cdot \cdot^2}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108</math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4</math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad</math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad</math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad</math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad</math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad</math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad</math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870062007 iTest Problems2017-08-12T15:49:14Z<p>Math piggy: /* Problem 11 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^2\cdot \cdot \cdot^2}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108</math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4</math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad</math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad</math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad</math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad</math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad</math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad</math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870052007 iTest Problems2017-08-12T15:48:14Z<p>Math piggy: /* Problem 11 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^2^\cdot^\cdot^\cdot^2}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108</math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4</math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad</math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad</math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad</math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad</math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad</math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad</math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870042007 iTest Problems2017-08-12T15:46:36Z<p>Math piggy: /* Problem 20 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108</math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4</math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad</math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad</math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad</math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad</math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad</math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad</math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870032007 iTest Problems2017-08-12T15:46:13Z<p>Math piggy: /* Problem 19 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108</math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4</math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad</math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad</math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad</math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad</math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870022007 iTest Problems2017-08-12T15:45:38Z<p>Math piggy: /* Problem 18 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108</math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4</math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad</math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad</math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870012007 iTest Problems2017-08-12T15:44:44Z<p>Math piggy: /* Problem 17 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=870002007 iTest Problems2017-08-12T15:44:08Z<p>Math piggy: /* Problem 15 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad</math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=869992007 iTest Problems2017-08-12T15:41:49Z<p>Math piggy: /* Problem 25 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad \\ </math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad</math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad</math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=869982007 iTest Problems2017-08-12T15:41:11Z<p>Math piggy: /* Problem 24 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad \\ </math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad</math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad</math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad \\ </math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad \\ </math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=869972007 iTest Problems2017-08-12T15:40:28Z<p>Math piggy: /* Problem 23 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad \\ </math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad</math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad</math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad</math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad \\ </math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad \\ </math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad \\ </math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad \\ </math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=869962007 iTest Problems2017-08-12T15:39:54Z<p>Math piggy: /* Problem 22 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad \\ </math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad</math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad \\ </math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad \\ </math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad \\ </math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad \\ </math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad \\ </math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad \\ </math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad \\ </math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=869952007 iTest Problems2017-08-12T15:39:13Z<p>Math piggy: /* Problem 21 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad \\ </math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad</math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad \\ </math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad \\ </math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad \\ </math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad \\ </math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad \\ </math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad \\ </math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad \\ </math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad \\ </math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=869942007 iTest Problems2017-08-12T15:38:42Z<p>Math piggy: /* Problem 21 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad \\ </math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad</math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad \\ </math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad \\ </math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad \\ </math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad \\ </math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad \\ </math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad \\ </math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad \\ </math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad \\ </math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad \\ </math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=869932007 iTest Problems2017-08-12T15:38:00Z<p>Math piggy: /* Problem 21 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad \\ </math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad \\ </math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad \\ </math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad \\ </math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad \\ </math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad \\ </math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad \\ </math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad \\ </math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad \\ </math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad \\ </math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
<br />
<br />
[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggyhttps://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems&diff=869922007 iTest Problems2017-08-12T15:37:23Z<p>Math piggy: /* Problem 43 */</p>
<hr />
<div>==Multiple Choice Section==<br />
<br />
===Problem 1===<br />
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? <br />
<br />
<math>\mathrm{(A)}\, 4</math><br />
<br />
[[2007 iTest Problems/Problem 1|Solution]]<br />
<br />
===Problem 2===<br />
Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy<br />
<math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.<br />
<br />
<math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math><br />
<br />
[[2007 iTest Problems/Problem 2|Solution]]<br />
<br />
===Problem 3===<br />
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?<br />
<br />
<math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math><br />
<br />
[[2007 iTest Problems/Problem 3|Solution]]<br />
<br />
===Problem 4===<br />
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.<br />
<br />
<math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 4|Solution]]<br />
<br />
===Problem 5===<br />
Compute the sum of all twenty-one terms of the geometric series<br />
<cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.<br />
<br />
<math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math><br />
<br />
[[2007 iTest Problems/Problem 5|Solution]]<br />
===Problem 6===<br />
Find the units digit of the sum<br />
<br />
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
<br />
[[2007 iTest Problems/Problem 6|Solution]]<br />
===Problem 7===<br />
An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.<br />
Find <math>s</math>.<br />
<br />
<math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math><br />
<br />
[[2007 iTest Problems/Problem 7|Solution]]<br />
===Problem 8===<br />
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?<br />
<br />
<math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math><br />
<br />
[[2007 iTest Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?<br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math><br />
<br />
[[2007 iTest Problems/Problem 10|Solution]]<br />
<br />
===Problem 11===<br />
Consider the "tower of power" <math>2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }2\qquad<br />
\text{(D) }3\qquad<br />
\text{(E) }4\qquad<br />
\text{(F) }5\qquad<br />
\text{(G) }6\qquad<br />
\text{(H) }7\qquad<br />
\text{(I) }8\qquad<br />
\text{(J) }9\qquad<br />
\text{(K) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 11|Solution]]<br />
<br />
===Problem 12===<br />
<br />
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.<br />
The game ends after one of the two teams scores three points (total, not necessarily consecutive). <br />
If every possible sequence of scores is equally likely, what is the expected score of the losing team.<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }1\qquad<br />
\text{(C) }\frac{3}{2}\qquad<br />
\text{(D) }\frac{8}{5}\qquad<br />
\text{(E) }\frac{5}{8}\qquad<br />
\text{(F) }2\qquad\\ \\<br />
\text{(G) }0\qquad<br />
\text{(H) }\frac{5}{2}\qquad<br />
\text{(I) }\frac{2}{5}\qquad<br />
\text{(J) }\frac{3}{4}\qquad<br />
\text{(K) }\frac{4}{3}\qquad<br />
\text{(L) }2007\qquad</math><br />
<br />
[[2007 iTest Problems/Problem 12|Solution]]<br />
<br />
===Problem 13===<br />
<br />
What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?<br />
<br />
<math>\text{(A) } 3 \quad<br />
\text{(B) } 4 \quad<br />
\text{(C) } 5 \quad<br />
\text{(D) } 6 \quad<br />
\text{(E) } 7 \quad<br />
\text{(F) } 8 \quad<br />
\text{(G) } 9 \quad<br />
\text{(H) } 10 \quad<br />
\text{(I) } 11 \quad<br />
\text{(J) } 12 \quad<br />
\text{(K) } 13 \quad<br />
\text{(L) } 14 \quad<br />
\text{(M) } 2007\quad </math><br />
<br />
[[2007 iTest Problems/Problem 13|Solution]]<br />
<br />
===Problem 14===<br />
<br />
Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12\quad<br />
\text{(N) } 13\quad </math><br />
<br />
[[2007 iTest Problems/Problem 14|Solution]]<br />
<br />
===Problem 15===<br />
<br />
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and<br />
placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the<br />
pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the<br />
equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?<br />
<br />
<math>\text{(A) }\frac{2}{3}\qquad<br />
\text{(B) }\frac{3}{4}\qquad<br />
\text{(C) }1\qquad<br />
\text{(D) }\frac{5}{4}\qquad<br />
\text{(E) }\frac{4}{3}\qquad<br />
\text{(F) }\frac{\sqrt{2}}{2}\qquad<br />
\text{(G) }\frac{\sqrt{3}}{2}\qquad<br />
\text{(H) }\sqrt{2}\qquad \\ </math><br />
<br />
<math>\text{(I) }\sqrt{3}\qquad<br />
\text{(J) }\frac{1+\sqrt{3}}{2}\qquad<br />
\text{(K) }\frac{2+\sqrt{6}}{2}\qquad<br />
\text{(L) }\frac{7}{6}\qquad<br />
\text{(M) }\frac{2+\sqrt{6}}{4}\qquad<br />
\text{(N) }\frac{4}{5}\qquad<br />
\text{(O) }2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 15|Solution]]<br />
<br />
===Problem 16===<br />
How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation<br />
<cmath>x^2+y^2=100</cmath><br />
<br />
<math>\text{(A) }1\qquad<br />
\text{(B) }2\qquad<br />
\text{(C) }4\qquad<br />
\text{(D) }5\qquad<br />
\text{(E) }41\qquad<br />
\text{(F) }42\qquad<br />
\text{(G) }69\qquad<br />
\text{(H) }76\qquad<br />
\text{(I) }130\qquad \\<br />
\text{(J) }133\qquad<br />
\text{(K) }233\qquad<br />
\text{(L) }311\qquad<br />
\text{(M) }317\qquad<br />
\text{(N) }420\qquad<br />
\text{(O) }520\qquad<br />
\text{(P) }2007</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 16|Solution]]<br />
<br />
===Problem 17===<br />
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.<br />
<br />
<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad<br />
\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad<br />
\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad<br />
\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad<br />
\text{(E) }1\qquad<br />
\text{(F) }\frac{5}{7}\qquad<br />
\text{(G) }\frac{3}{7}\qquad<br />
\text{(H) }6\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{6}\qquad<br />
\text{(J) }\frac{1}{2}\qquad<br />
\text{(K) }\frac{6}{7}\qquad<br />
\text{(L) }\frac{4}{7}\qquad<br />
\text{(M) }\sqrt{3}\qquad<br />
\text{(N) }\frac{\sqrt{3}}{3}\qquad<br />
\text{(O) }\frac{5}{6}\qquad<br />
\text{(P) }\frac{2}{3}\qquad <br />
\text{(Q) }\frac{1}{2007}\qquad\\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 17|Solution]]<br />
<br />
===Problem 18===<br />
Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.<br />
If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }4\qquad<br />
\text{(C) }108\qquad<br />
\text{(D) It could be }0 \text{ or } 4\qquad<br />
\text{(E) It could be }0 \text{ or } 108 \\ </math><br />
<br />
<math>\text{(F) }18\qquad<br />
\text{(G) }-4\qquad<br />
\text{(H) }-108\qquad<br />
\text{(I) It could be } 0 \text{ or } -4 \\ </math><br />
<br />
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad<br />
\text{(K) It could be } 4 \text{ or } {-4}\qquad<br />
\text{(L) There is no such value of } r\qquad \\ </math><br />
<br />
<math>\text{(M) } 1 \qquad<br />
\text{(N) } {-2} \qquad <br />
\text{(O) It could be } 4 \text{ or } -4 \qquad<br />
\text{(P) It could be } 0 \text{ or } -2 \qquad \\ </math><br />
<br />
<math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad<br />
\text{(R) } 2007 \\ </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 18|Solution]]<br />
<br />
===Problem 19===<br />
<br />
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"<br />
Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.<br />
If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?<br />
<br />
<math>\textbf{(A) }0\qquad<br />
\textbf{(B) }\dfrac1{10}\qquad<br />
\textbf{(C) }\dfrac18\qquad<br />
\textbf{(D) }\dfrac15\qquad<br />
\textbf{(E) }\dfrac14\qquad<br />
\textbf{(F) }\dfrac13\qquad<br />
\textbf{(G) }\dfrac25\qquad \\ </math><br />
<math>\textbf{(H) }\dfrac12\qquad<br />
\textbf{(I) }\dfrac35\qquad<br />
\textbf{(J) }\dfrac23\qquad<br />
\textbf{(K) }\dfrac45\qquad<br />
\textbf{(L) }1\qquad<br />
\textbf{(M) }\dfrac54\qquad \\ </math><br />
<math>\textbf{(N) }\dfrac43\qquad<br />
\textbf{(O) }\dfrac32\qquad<br />
\textbf{(P) }2\qquad<br />
\textbf{(Q) }3\qquad<br />
\textbf{(R) }4\qquad<br />
\textbf{(S) }2007 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 19|Solution]]<br />
<br />
===Problem 20===<br />
<br />
Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math><br />
<br />
<math>\text{(A) } 1\qquad<br />
\text{(B) } 2\qquad<br />
\text{(C) } 3\qquad<br />
\text{(D) } 4\qquad<br />
\text{(E) } 5\qquad<br />
\text{(F) } 6\qquad<br />
\text{(G) } 7\qquad<br />
\text{(H) } 8\qquad \\ </math><br />
<math>\text{(I) } 9\qquad<br />
\text{(J) } 10\qquad<br />
\text{(K) } 11\qquad<br />
\text{(L) } 12\qquad<br />
\text{(M) } 13\qquad<br />
\text{(N) } 14\qquad<br />
\text{(O) } 15\qquad<br />
\text{(P) } 16\qquad \\ </math><br />
<math>\text{(Q) } 55\qquad<br />
\text{(R) } 63\qquad<br />
\text{(S) } 64\qquad<br />
\text{(T) } 2007\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 20|Solution]]<br />
<br />
===Problem 21===<br />
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.<br />
One such example is <br />
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath><br />
What is the probability that the value of the expression James wrote down is <math>7 \\ </math>?<br />
<br />
<math>\text{(A) }0\qquad<br />
\text{(B) }\frac{6435 }{2^{14}}\qquad<br />
\text{(C) }\frac{6435 }{2^{13}}\qquad<br />
\text{(D) }\frac{429}{2^{12}}\qquad<br />
\text{(E) }\frac{429}{2^{11}}\qquad<br />
\text{(F) }\frac{429}{2^{10}}\qquad<br />
\text{(G) }\frac{1}{15}\qquad<br />
\text{(H) }\frac{1}{31}\qquad \\ </math><br />
<br />
<math>\text{(I) }\frac{1}{30}\qquad<br />
\text{(J) }\frac{1}{29}\qquad<br />
\text{(K) }\frac{6435 }{2^{15}}\qquad<br />
\text{(L) }\frac{6435 }{2^{14}}\qquad<br />
\text{(M) }\frac{6435 }{2^{13}}\qquad<br />
\text{(N) }\frac{1}{2^{7}}\qquad<br />
\text{(O) }\frac{1}{2^{14}}\qquad<br />
\text{(P) }\frac{1}{2^{15}}\qquad \\ </math><br />
<br />
<math>\text{(P) }\frac{2007}{2^{14}}\qquad<br />
\text{(P) }\frac{2007}{2^{15}}\qquad<br />
\text{(P) }\frac{2007}{2^{2007}}\qquad<br />
\text{(P) }\frac{1}{2007}\qquad<br />
\text{(P) }\frac{-2007}{2^{14}}\qquad</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 21|Solution]]<br />
<br />
===Problem 22===<br />
Find the value of <math>c</math> such that the system of equations<br />
<cmath> |x+y|=2007 \\ |x-y|=c</cmath><br />
has exactly two solutions <math>(x,y)</math> in real numbers.<br />
<br />
<br />
<math>\text{(A) } 0 \quad<br />
\text{(B) } 1 \quad<br />
\text{(C) } 2 \quad<br />
\text{(D) } 3 \quad<br />
\text{(E) } 4 \quad<br />
\text{(F) } 5 \quad<br />
\text{(G) } 6 \quad<br />
\text{(H) } 7 \quad<br />
\text{(I) } 8 \quad<br />
\text{(J) } 9 \quad<br />
\text{(K) } 10 \quad<br />
\text{(L) } 11 \quad<br />
\text{(M) } 12 \quad \\ </math><br />
<br />
<math>\text{(N) } 13 \quad<br />
\text{(O) } 14 \quad<br />
\text{(P) } 15 \quad<br />
\text{(Q) } 16 \quad<br />
\text{(R) } 17 \quad<br />
\text{(S) } 18 \quad<br />
\text{(T) } 223 \quad<br />
\text{(U) } 678 \quad<br />
\text{(V) } 2007 \quad </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 22|Solution]]<br />
<br />
===Problem 23===<br />
Find the product of the non-real roots of the equation<br />
<cmath>(x^2-3)^2+5(x^2-3)+6=0</cmath><br />
<br />
<math>\text{(A) } 0\quad<br />
\text{(B) } 1\quad<br />
\text{(C) } -1\quad<br />
\text{(D) } 2\quad<br />
\text{(E) } -2\quad<br />
\text{(F) } 3\quad<br />
\text{(G) } -3\quad<br />
\text{(H) } 4\quad<br />
\text{(I) } -4\quad \\ </math><br />
<br />
<math>\text{(J) } 5\quad<br />
\text{(K) } -5\quad<br />
\text{(L) } 6\quad<br />
\text{(M) } -6\quad<br />
\text{(N) } 3+2i\quad<br />
\text{(O) } 3-2i\quad \\ </math><br />
<br />
<math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad<br />
\text{(Q) } 8\quad <br />
\text{(R) } -8\qquad<br />
\text{(S) } 12\quad<br />
\text{(T) } -12\quad<br />
\text{(U) } 42\quad \\ </math><br />
<br />
<math>\text{(V) Ying} \quad<br />
\text{(W) } 207</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 23|Solution]]<br />
<br />
===Problem 24===<br />
Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.<br />
Find the remainder when <math>N</math> is divided by <math>25</math>.<br />
<br />
<math>\text{(A) }0 \quad<br />
\text{(B) }1 \quad<br />
\text{(C) }2 \quad<br />
\text{(D) }3 \quad<br />
\text{(E) }4 \quad<br />
\text{(F) }5 \quad<br />
\text{(G) }6 \quad<br />
\text{(H) }7 \quad<br />
\text{(I) } 8\quad \\ </math><br />
<br />
<math>\text{(J) }9 \quad<br />
\text{(K) }10 \quad<br />
\text{(L) }11 \quad<br />
\text{(M) }12 \quad<br />
\text{(N) }13 \quad<br />
\text{(O) }14 \quad<br />
\text{(P) }15 \quad<br />
\text{(Q) }16 \quad \\ </math><br />
<br />
<math>\text{(R) }17 \quad<br />
\text{(S) }18 \quad<br />
\text{(T) }19 \quad<br />
\text{(U) }20 \quad<br />
\text{(V) }21 \quad<br />
\text{(W) }22 \quad<br />
\text{(X) }23 </math><br />
<br />
<br />
[[2007 iTest Problems/Problem 24|Solution]]<br />
<br />
===Problem 25===<br />
<br />
Ted's favorite number is equal to<br />
<cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath><br />
<br />
Find the remainder when Ted's favorite number is divided by 25.<br />
<br />
<math>\text{(A) } 0\qquad<br />
\text{(B) } 1\qquad<br />
\text{(C) } 2\qquad<br />
\text{(D) } 3\qquad<br />
\text{(E) } 4\qquad<br />
\text{(F) } 5\qquad<br />
\text{(G) } 6\qquad<br />
\text{(H) } 7\qquad<br />
\text{(I) } 8\qquad \\ </math><br />
<br />
<math>\text{(J) } 9\qquad<br />
\text{(K) } 10\qquad<br />
\text{(L) } 11\qquad<br />
\text{(M) } 12\qquad<br />
\text{(N) } 13\qquad<br />
\text{(O) } 14\qquad<br />
\text{(P) } 15\qquad<br />
\text{(Q) } 16\qquad \\ </math><br />
<br />
<math>\text{(R) } 17\qquad<br />
\text{(S) } 18\qquad<br />
\text{(T) } 19\qquad<br />
\text{(U) } 20\qquad<br />
\text{(V) } 21\qquad<br />
\text{(A) } 22\qquad<br />
\text{(X) } 23\qquad<br />
\text{(Y) } 24</math><br />
<br />
<br />
[[2007 iTest Problems/Problem 25|Solution]]<br />
<br />
==Short Answer Section==<br />
<br />
===Problem 26===<br />
<br />
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of \$370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of \$180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday? <br />
<br />
<br />
[[2007 iTest Problems/Problem 26|Solution]]<br />
<br />
===Problem 27===<br />
<br />
The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube.<br />
<br />
<br />
[[2007 iTest Problems/Problem 27|Solution]]<br />
<br />
===Problem 28===<br />
<br />
The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. <br />
<br />
<br />
[[2007 iTest Problems/Problem 28|Solution]]<br />
<br />
===Problem 29===<br />
<br />
Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. <br />
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[[2007 iTest Problems/Problem 29|Solution]]<br />
<br />
===Problem 30===<br />
<br />
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>. <br />
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[[2007 iTest Problems/Problem 30|Solution]]<br />
<br />
===Problem 31===<br />
<br />
Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{integral value}</math> of the area of the triangle. <br />
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[[2007 iTest Problems/Problem 31|Solution]]<br />
<br />
===Problem 32===<br />
<br />
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio <math>2</math> to <math>1</math>. How many integer values of k are there such that <math>0<k\leq 2007</math> and the area between the parabola <math>y=k-x^2</math> and the <math>x</math>-axis is an integer?<br />
<br />
<asy><br />
import graph;<br />
size(300);<br />
defaultpen(linewidth(0.8)+fontsize(10));<br />
real k=1.5;<br />
real endp=sqrt(k);<br />
real f(real x) {<br />
return k-x^2;<br />
}<br />
path parabola=graph(f,-endp,endp)--cycle;<br />
filldraw(parabola, lightgray);<br />
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));<br />
label("Region I", (0,2*k/5));<br />
label("Box II", (51/64*endp,13/16*k));<br />
label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));</asy><br />
<br />
<br />
[[2007 iTest Problems/Problem 32|Solution]]<br />
<br />
===Problem 33===<br />
<br />
How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? <br />
<br />
<br />
[[2007 iTest Problems/Problem 33|Solution]]<br />
<br />
===Problem 34===<br />
<br />
Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b</math>. <br />
<br />
<br />
[[2007 iTest Problems/Problem 34|Solution]]<br />
<br />
===Problem 35===<br />
<br />
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 35|Solution]]<br />
<br />
===Problem 36===<br />
<br />
Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. <br />
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<br />
[[2007 iTest Problems/Problem 36|Solution]]<br />
<br />
===Problem 37===<br />
<br />
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are <math>60, 20</math>, and <math>15</math> respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area. <br />
<br />
<br />
[[2007 iTest Problems/Problem 37|Solution]]<br />
<br />
===Problem 38===<br />
<br />
Find the largest positive integer that is equal to the cube of the sum of its digits. <br />
<br />
<br />
[[2007 iTest Problems/Problem 38|Solution]]<br />
<br />
===Problem 39===<br />
<br />
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}</math>. Find <math>a+b</math>. <br />
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<br />
[[2007 iTest Problems/Problem 39|Solution]]<br />
<br />
===Problem 40===<br />
<br />
Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. <br />
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<br />
[[2007 iTest Problems/Problem 40|Solution]]<br />
<br />
===Problem 41===<br />
<br />
The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define <math>f(n)</math> to be <math>m</math>. For example, <math>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value of <math>f(2007)</math>. <br />
<br />
[[2007 iTest Problems/Problem 41|Solution]]<br />
<br />
===Problem 42===<br />
<br />
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as <math>\dfrac{a-b\sqrt c}d</math>, where all four variables are positive integers, <math>c</math> is a multple of no perfect square greater than <math>1</math>, a is coprime with <math>d</math>, and <math>b</math> is coprime with <math>d</math>. Find the value of <math>a+b+c+d</math>. <br />
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[[2007 iTest Problems/Problem 42|Solution]]<br />
<br />
===Problem 43===<br />
<br />
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following <math>100</math> <math>9</math>-digit integers: <br />
<br />
<cmath>\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} </cmath><br />
<br />
She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. <br />
<br />
[[2007 iTest Problems/Problem 43|Solution]]<br />
<br />
===Problem 44===<br />
<br />
A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are natural numbers such that <math>a/b</math> is the probability that <math>N</math> and <math>n^3-36n</math> are relatively prime, find the value of <math>a+b</math>.<br />
<br />
[[2007 iTest Problems/Problem 44|Solution]]<br />
<br />
===Problem 45===<br />
<br />
Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> digits, <math>a\neq 0</math>. <br />
<br />
[[2007 iTest Problems/Problem 45|Solution]]<br />
<br />
===Problem 46===<br />
<br />
Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <br />
<cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*}</cmath><br />
If <math>m</math> is the minimum possible value of <math>\lfloor x^3+y^3+z^3\rfloor</math>, find the modulo <math>2007</math> residue of <math>m</math>.<br />
<br />
[[2007 iTest Problems/Problem 46|Solution]]<br />
<br />
===Problem 47===<br />
<br />
Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <br />
<br />
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\<br />
Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath><br />
<br />
Let <math>k</math> be the largest integer that satisfies all of the following conditions: <math>|X_i-k|\leq 2007</math>, for some positive integer <math>i</math>;<br />
<math>|Y_j-k|\leq 2007</math>, for some positive integer <math>j</math>; and<br />
<math>k<10^{2007}</math>.<br />
Find the remainder when <math>k</math> is divided by <math>2007</math>. <br />
<br />
[[2007 iTest Problems/Problem 47|Solution]]<br />
<br />
===Problem 48===<br />
<br />
Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}</math>, where, for <math>1\leq i\leq 2007, x_i</math> is a nonnegative real number, and <math>x_1+x_2+x_3+\cdots+x_{2007}=\pi</math>. Find the value of <math>a+b</math>. <br />
<br />
[[2007 iTest Problems/Problem 48|Solution]]<br />
<br />
===Problem 49===<br />
<br />
How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? <br />
<br />
[[2007 iTest Problems/Problem 49|Solution]]<br />
<br />
===Problem 50===<br />
<br />
A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder with radius <math>10</math> and height <math>3</math> so that the centers of the square and circle coincide. If <math>V</math> is the smallest convex region that contains <math>Z</math>, calculate <math>\lfloor\operatorname{vol}V\rfloor</math> (the greatest integer less than or equal to the volume of <math>V</math>). <br />
<br />
[[2007 iTest Problems/Problem 50|Solution]]<br />
<br />
==Ultimate Question==<br />
<br />
===Problem 51===<br />
Find the highest point (largest possible <math>y</math>-coordinate) on the parabola<br />
<cmath>y=-2x^2+ 28x+ 418</cmath><br />
<br />
===Problem 52===<br />
Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both<br />
<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.<br />
<br />
===Problem 53===<br />
Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be the smallest possible value of their sum. Find the remainder when <math>N</math> is divided by <math>2007</math>. <br />
<br />
===Problem 54===<br />
Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them, we get the sequence <math>(1, 2006, 2007)</math>. Repeating the process of inserting differences between numbers, we get the sequence <math>(1, 2005, 2006, 1, 2007)</math>. A third iteration of this process results in <math>(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>4T</math> (that is, <math>4</math> times the integer <math>T</math>) appears a total of <math>N</math> times among these <math>2^{2007}+1</math> terms, find the remainder when <math>N</math> gets divided by <math>2007</math>. <br />
===Problem 55===<br />
Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the value of <math>\lfloor x\rfloor</math>. <br />
===Problem 56===<br />
Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right of the radix point are <math>0</math>'s? <br />
===Problem 57===<br />
Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>0^2=0</math> is considered a perfect square.) <br />
===Problem 58===<br />
Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define<br />
<cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath><br />
Compute the value of <math>S(10,T+55)-S(10,55)+S(10,T-55)</math>.<br />
<br />
===Problem 59===<br />
Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-332)/(2T-601)</math>. Find the value of <math>a</math>. <br />
===Problem 60===<br />
Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>. <br />
== Tiebreaker Questions ==<br />
<br />
=== Problem TB1 ===<br />
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.<br />
<br />
[[2007 iTest Problems/Problem TB1|Solution]]<br />
<br />
=== Problem TB2 ===<br />
Factor completely over integer coefficients the polynomial <math>p(x)=x^8+x^5+x^4+x^3+x+1</math>. Demonstrate that your factorization is complete.<br />
<br />
[[2007 iTest Problems/Problem TB2|Solution]]<br />
<br />
=== Problem TB3 ===<br />
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.<br />
<br />
[[2007 iTest Problems/Problem TB3|Solution]]<br />
<br />
=== Problem TB4 ===<br />
Circle <math>O</math> is the circumcircle of non-isosceles triangle <math>ABC</math>. The tangent lines to circle <math>O</math> at points <math>B</math> and <math>C</math> intersect at <math>L_a</math>, and the tangents at <math>A</math> and <math>C</math> intersect at <math>L_b</math>. The external angle bisectors of triangle <math>ABC</math> at <math>B</math> and <math>C</math> meet at <math>I_a</math> and the external bisectors at <math>A</math> and <math>C</math> intersect at <math>I_b</math>. Prove that lines <math>L_aI_a</math>, <math>L_bI_b</math>, and <math>AB</math> are concurrent.<br />
<br />
[[2007 iTest Problems/Problem TB4|Solution]]<br />
<br />
<br />
{{stub fart yo}}</div>Math piggy