https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=YottaByte&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T06:31:49ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12A_Problems/Problem_2&diff=345332005 AMC 12A Problems/Problem 22010-05-17T23:07:33Z<p>YottaByte: Said 12A, had me worried I had been doing AMC 12 problems.</p>
<hr />
<div>{{Duplicate|[[2005 AMC 10A Problems|2005 AMC 10A #2]] and [[2005 AMC 10A Problems|2005 AMC 10A #3]]}}<br />
<br />
== Problem ==<br />
The equations <math>2x + 7 = 3</math> and <math>bx - 10 = - 2</math> have the same solution. What is the value of <math>b</math>?<br />
<br />
<math><br />
(\mathrm {A}) \ -8 \qquad (\mathrm {B}) \ -4 \qquad (\mathrm {C})\ 2 \qquad (\mathrm {D}) \ 4 \qquad (\mathrm {E})\ 8<br />
</math><br />
<br />
== Solution ==<br />
<math>2x + 7 = 3 \Longrightarrow x = -2, \quad -2b - 10 = -2 \Longrightarrow -2b = 8 \Longrightarrow b = -4\ \mathrm{(B)}</math><br />
<br />
<br />
== See also ==<br />
{{AMC12 box|year=2005|num-b=1|num-a=3|ab=A}}<br />
{{AMC10 box|year=2005|ab=A|num-b=2|num-a=4}}<br />
[[Category:Introductory Algebra Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=1966_AHSME_Problems&diff=342841966 AHSME Problems2010-04-12T11:59:38Z<p>YottaByte: /* Problem 5 */</p>
<hr />
<div>== Problem 1 ==<br />
Given that the ratio of <math>3x - 4</math> to <math>y + 15</math> is constant, and <math>y = 3</math> when <math>x = 2</math>, then, when <math>y = 12</math>, <math>x</math> equals:<br />
<br />
<math>\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qquad \text{(D)} \ \frac72 \qquad \text{(E)} \ 8</math><br />
<br />
[[1966 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
If the arithmetic mean of two numbers is <math>6</math> and their geometric mean is <math>10</math>, then an equation with the given two numbers as roots is:<br />
<br />
<math>\text{(A)} \ x^2 + 12x + 100 = 0 ~~ \text{(B)} \ x^2 + 6x + 100 = 0 ~~ \text{(C)} \ x^2 - 12x - 10 = 0</math><br />
<math>\text{(D)} \ x^2 - 12x + 100 = 0 \qquad \text{(E)} \ x^2 - 6x + 100 = 0</math><br />
<br />
[[1966 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
Circle I is circumscribed about a given square and circle II is inscribed in the given square. If <math>r</math> is the ratio of the area of circle <math>I</math> to that of circle <math>II</math>, then <math>r</math> equals:<br />
<br />
<math>\text{(A)} \sqrt 2 \qquad \text{(B)} 2 \qquad \text{(C)} \sqrt 3 \qquad \text{(D)} 2\sqrt 2 \qquad \text{(E)} 2\sqrt 3</math><br />
<br />
[[1966 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
The number of values of <math>x</math> satisfying the equation<br />
<br />
<math>\frac {2x^2 - 10x}{x^2 - 5x} = x - 3</math><br />
<br />
is:<br />
<br />
<math>\text{(A)} \ \text{zero} \qquad \text{(B)} \ \text{one} \qquad \text{(C)} \ \text{two} \qquad \text{(D)} \ \text{three} \qquad \text{(E)} \ \text{an integer greater than 3}</math><br />
<br />
[[1966 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<math>AB</math> is the diameter of a circle centered at <math>O</math>. <math>C</math> is a point on the circle such that angle <math>BOC</math> is <math>60^\circ</math>. If the diameter of the circle is <math>5</math> inches, the length of chord <math>AC</math>, expressed in inches, is:<br />
<br />
<math>\text{(A)} \ 3 \qquad \text{(B)} \ \frac {5\sqrt {2}}{2} \qquad \text{(C)} \frac {5\sqrt3}{2} \ \qquad \text{(D)} \ 3\sqrt3 \qquad \text{(E)} \ \text{none of these}</math><br />
<br />
[[1966 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
Let <math>\frac {35x - 29}{x^2 - 3x + 2} = \frac {N_1}{x - 1} + \frac {N_2}{x - 2}</math> be an identity in <math>x</math>. The numerical value of <math>N_1N_2</math> is:<br />
<br />
<math>\text{(A)} \ - 246 \qquad \text{(B)} \ - 210 \qquad \text{(C)} \ - 29 \qquad \text{(D)} \ 210 \qquad \text{(E)} \ 246</math><br />
<br />
[[1966 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
The length of the common chord of two intersecting circles is <math>16</math> feet. If the radii are <math>10</math> feet and <math>17</math> feet, a possible value for the distance between the centers of teh circles, expressed in feet, is:<br />
<br />
<math>\text{(A)} \ 27 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ \sqrt {389} \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{undetermined}</math><br />
<br />
[[1966 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
If <math>x = (\log_82)^{(\log_28)}</math>, then <math>\log_3x</math> equals:<br />
<br />
<math>\text{(A)} \ - 3 \qquad \text{(B)} \ - \frac13 \qquad \text{(C)} \ \frac13 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 9</math><br />
<br />
[[1966 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is:<br />
<br />
<math>\text{(A)} \ 2 \qquad \text{(B)} \ - 2 - \frac {3i\sqrt {3}}{4} \qquad \text{(C)} \ 0 \qquad \text{(D)} \ - \frac {3i\sqrt {3}}{4} \qquad \text{(E)} \ - 2</math><br />
<br />
[[1966 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
The sides of triangle <math>BAC</math> are in the ratio <math>2: 3: 4</math>. <math>BD</math> is the angle-bisector drawn to the shortest side <math>AC</math>, dividing it into segments <math>AD</math> and <math>CD</math>. If the length of <math>AC</math> is <math>10</math>, then the length of the longer segment of <math>AC</math> is:<br />
<br />
<math>\text{(A)} \ 3\frac12 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 5\frac57 \qquad \text{(D)} \ 6 \qquad \text{(E)} \ 7\frac12</math><br />
<br />
[[1966 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Five points <math>O,A,B,C,D</math> are taken in order on a straight line with distances <math>OA = a</math>, <math>OB = b</math>, <math>OC = c</math>, and <math>OD = d</math>. <math>P</math> is a point on the line between <math>B</math> and <math>C</math> and such that <math>AP: PD = BP: PC</math>. Then <math>OP</math> equals:<br />
<br />
<math>\textbf{(A)} \frac {b^2 - bc}{a - b + c - d} \qquad \textbf{(B)} \frac {ac - bd}{a - b + c - d} \\<br />
\textbf{(C)} - \frac {bd + ac}{a - b + c - d} \qquad \textbf{(D)} \frac {bc + ad}{a + b + c + d} \qquad \textbf{(E)} \frac {ac - bd}{a + b + c + d}</math><br />
<br />
[[1966 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
<br />
<br />
[[1966 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[1966 AHSME]]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2000_AMC_8_Problems/Problem_1&diff=342432000 AMC 8 Problems/Problem 12010-04-06T21:12:25Z<p>YottaByte: Created page with '== Problem == Aunt Anna is <math>42</math> years old. Caitlin is <math>5</math> years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? <math> \…'</p>
<hr />
<div>== Problem ==<br />
Aunt Anna is <math>42</math> years old. Caitlin is <math>5</math> years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?<br />
<br />
<math><br />
\mathrm{(A)}\ 15<br />
\qquad<br />
\mathrm{(B)}\ 16<br />
\qquad<br />
\mathrm{(C)}\ 17<br />
\qquad<br />
\mathrm{(D)}\ 21<br />
\qquad<br />
\mathrm{(E)}\ 37<br />
</math><br />
== Solution ==<br />
If Brianna is half as old as Aunt Anna, then Briana is <math>42/2</math> years old, or <math>21</math> years old. <br />
<br />
If Caitlin is <math>5</math> years younger than Briana, she is <math>21-5</math> years old, or <math>16</math>.<br />
<br />
So, the answer is <math>\boxed{B}</math></div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=1993_AJHSME_Problems/Problem_2&diff=340921993 AJHSME Problems/Problem 22010-04-01T20:28:34Z<p>YottaByte: Created page with '== Problem == When the fraction <math>\dfrac{49}{84}</math> is expressed in simplest form, then the sum of the numerator and the denominator will be <math>\text{(A)}\ 11 \qquad…'</p>
<hr />
<div>== Problem ==<br />
When the fraction <math>\dfrac{49}{84}</math> is expressed in simplest form, then the sum of the numerator and the denominator will be <br />
<br />
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 133</math><br />
<br />
== Solution == <br />
The fraction is already in simplest form.<br />
The prime factorization of <math>49</math> is <math>7^2</math> and for <math>84</math> it is <math>2*37</math>, so the greatest common factor is 1. <br />
<br />
Then we add <math>49+84</math> and get <math>133 \Rightarrow E</math>.</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_14&diff=340432007 AMC 8 Problems/Problem 142010-03-26T21:10:53Z<p>YottaByte: /* Solution */</p>
<hr />
<div>== Problem ==<br />
<br />
The base of isosceles <math>\triangle ABC</math> is <math>24</math> and its area is <math>60</math>. What is the length of one<br />
of the congruent sides?<br />
<br />
<math>\mathrm{(A)}\ 5 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 13 \qquad \mathrm{(D)}\ 14 \qquad \mathrm{(E)}\ 18</math><br />
<br />
== Solution ==<br />
<br />
The area of a triangle is shown by <math>\frac{1}{2}bh</math>.<br />
<br />
We set the base equal to <math>24</math>, and the area equal to <math>60</math>,<br />
<br />
and we get the height, or altitude, of the triangle to be <math>5</math>.<br />
<br />
In this isosceles triangle, the height bisects the base, <br />
<br />
so by using the pythagorean theorem, <math>a^2+b^2=c^2</math>,<br />
<br />
we can solve for one of the legs of the triangle (it will be the the hypotenuse, <math>c</math>).<br />
<br />
<math>a = 12</math>, <math>b = 5</math>,<br />
<br />
<math>c = 13</math><br />
<br />
The answer is <math>\boxed{C}</math></div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_14&diff=340422007 AMC 8 Problems/Problem 142010-03-26T21:10:00Z<p>YottaByte: /* Solution */</p>
<hr />
<div>== Problem ==<br />
<br />
The base of isosceles <math>\triangle ABC</math> is <math>24</math> and its area is <math>60</math>. What is the length of one<br />
of the congruent sides?<br />
<br />
<math>\mathrm{(A)}\ 5 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 13 \qquad \mathrm{(D)}\ 14 \qquad \mathrm{(E)}\ 18</math><br />
<br />
== Solution ==<br />
<br />
The area of a triangle is shown by <math>\frac{1}{2}bh</math>.<br />
<br />
We set the base equal to <math>24</math>, and the area equal to <math>60</math>,<br />
<br />
and we get the height, or altitude, of the triangle to be <math>5</math>.<br />
<br />
In this isosceles triangle, the height bisects the base, <br />
<br />
so by using the pythagorean theorem, <math>a^2+b^2=c^2</math>,<br />
<br />
we can solve for one of the legs of the triangle (it will be the the hypotenuse, <math>c</math>).<br />
<br />
<math>c = 13</math><br />
<br />
The answer is <math>\boxed{C}</math></div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_14&diff=340412007 AMC 8 Problems/Problem 142010-03-26T21:07:18Z<p>YottaByte: Created page with '== Problem == The base of isosceles <math>\triangle ABC</math> is <math>24</math> and its area is <math>60</math>. What is the length of one of the congruent sides? <math>\math…'</p>
<hr />
<div>== Problem ==<br />
<br />
The base of isosceles <math>\triangle ABC</math> is <math>24</math> and its area is <math>60</math>. What is the length of one<br />
of the congruent sides?<br />
<br />
<math>\mathrm{(A)}\ 5 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 13 \qquad \mathrm{(D)}\ 14 \qquad \mathrm{(E)}\ 18</math><br />
<br />
== Solution ==<br />
<br />
The area of a triangle is shown by <math>\frac{1}{2}bh</math>.<br />
<br />
We set the base equal to <math>24</math>, and the area equal to <math>60</math>,<br />
<br />
and we get the height, or altitude, of the triangle to be <math>5</math>.<br />
<br />
In this isosceles triangle, the height bisects the base, <br />
<br />
so by using the pythagorean theorem, <math>a^2+b^2=c^2</math>,<br />
<br />
we can solve for one of the legs of the triangle (it will be the the hypotenuse, <math>c</math>).<br />
<br />
&12^2+5^2=c^2<math><br />
<br />
</math>c = 13<math><br />
<br />
The answer is </math>\boxed{C}$</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_15&diff=340402007 AMC 8 Problems/Problem 152010-03-26T20:50:56Z<p>YottaByte: Created page with '== Problem == Let <math>a, b</math> and <math>c</math> be numbers with <math>0 < a < b < c</math>. Which of the following is impossible? <math>\mathrm{(A)} \ a + c < b \qquad …'</p>
<hr />
<div>== Problem ==<br />
<br />
Let <math>a, b</math> and <math>c</math> be numbers with <math>0 < a < b < c</math>. Which of the following is<br />
impossible?<br />
<br />
<math>\mathrm{(A)} \ a + c < b \qquad \mathrm{(B)} \ a * b < c \qquad \mathrm{(C)} \ a + b < c \qquad \mathrm{(D)} \ a * c < b \qquad \mathrm{(E)}\frac{b}{c} = a</math><br />
<br />
== Solution ==<br />
<br />
According to the given rules, <br />
<br />
Every number needs to be positive. <br />
<br />
Since <math>c</math> is always greater than <math>b</math>, <br />
<br />
adding a positive number (<math>a</math>) to <math>c</math> will always make it greater than <math>b</math>.<br />
<br />
Therefore, the answer is <math>\boxed{A}</math></div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_13&diff=340392007 AMC 8 Problems/Problem 132010-03-26T20:41:41Z<p>YottaByte: Created page with '== Problem == Sets <math>A</math> and <math>B</math>, shown in the Venn diagram, have the same number of elements. Their union has <math>2007</math> elements and their intersect…'</p>
<hr />
<div>== Problem ==<br />
<br />
Sets <math>A</math> and <math>B</math>, shown in the Venn diagram, have the same number of elements.<br />
Their union has <math>2007</math> elements and their intersection has <math>1001</math> elements. Find<br />
the number of elements in <math>A</math>.<br />
<br />
<center>[[Image:AMC8_2007_13.png]]</center><br />
<br />
<math>\mathrm{(A)}\ 503 \qquad \mathrm{(B)}\ 1006 \qquad \mathrm{(C)}\ 1504 \qquad \mathrm{(D)}\ 1507 \qquad \mathrm{(E)}\ 1510</math><br />
<br />
== Solution ==<br />
<br />
Let <math>x</math> be the number of elements in <math>A</math> and <math>B</math>. <br />
<br />
Since the union is the sum of all elements in <math>A</math> and <math>B</math>, <br />
<br />
and <math>A</math> and <math>B</math> have the same number of elements then,<br />
<br />
<math>2x-1001 = 2007</math><br />
<br />
<math>2x = 3008</math><br />
<br />
<math>x = 1504</math>. <br />
<br />
The answer is <math>\boxed{C}</math></div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2002_AMC_10B_Problems&diff=340382002 AMC 10B Problems2010-03-26T18:16:19Z<p>YottaByte: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
The ratio <math>\frac{2^{2001}\cdot3^{2003}}{6^{2002}}</math> is:<br />
<br />
<math> \mathrm{(A) \ } 1/6\qquad \mathrm{(B) \ } 1/3\qquad \mathrm{(C) \ } 1/2\qquad \mathrm{(D) \ } 2/3\qquad \mathrm{(E) \ } 3/2 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
For the nonzero numbers a, b, and c, define<br />
<br />
<math>(a,b,c)=\frac{abc}{a+b+c}</math><br />
<br />
Find <math>(2,4,6)</math>.<br />
<br />
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 24 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The arithmetic mean of the nine numbers in the set <math>\{9,99,999,9999,\ldots,999999999\}</math> is a 9-digit number <math>M</math>, all of whose digits are distinct. The number <math>M</math> does not contain the digit<br />
<br />
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 8 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
What is the value of<br />
<br />
<math>(3x-2)(4x+1)-(3x-2)4x+1</math><br />
<br />
when <math>x=4</math>?<br />
<br />
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
For how many positive integers n is <math>n^2-3n+2</math> a prime number?<br />
<br />
<math> \mathrm{(A) \ } \text{none}\qquad \mathrm{(B) \ } \text{one}\qquad \mathrm{(C) \ } \text{two}\qquad \mathrm{(D) \ } \text{more than two, but finitely many}\qquad \mathrm{(E) \ } \text{infinitely many} </math><br />
<br />
[[2002 AMC 10B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
Let <math>n</math> be a positive integer such that <math>\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}</math> is an integer. Which of the following statements is '''not''' true?<br />
<br />
<math> \mathrm{(A) \ } 2\text{ divides }n\qquad \mathrm{(B) \ } 3\text{ divides }n\qquad \mathrm{(C) \ } 6\text{ divides }n\qquad \mathrm{(D) \ } 7\text{ divides }n\qquad \mathrm{(E) \ } n>84 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
<br />
Suppose July of year <math>N</math> has five Mondays. Which of the following must occurs five times in the August of year <math>N</math>? (Note: Both months have <math>31</math> days.)<br />
<br />
<math>\textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Thursday} \qquad \textbf{(E)}\ \text{Friday}</math><br />
<br />
[[2002 AMC 10B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
<br />
Using the letters <math>A</math>, <math>M</math>, <math>O</math>, <math>S</math>, and <math>U</math>, we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" <math>USAMO</math> occupies position<br />
<br />
<math> \mathrm{(A) \ } 112\qquad \mathrm{(B) \ } 113\qquad \mathrm{(C) \ } 114\qquad \mathrm{(D) \ } 115\qquad \mathrm{(E) \ } 116 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
Suppose that <math>a</math> and <math>b</math> are nonzero real numbers, and that the equation <math>x^2+ax+b=0</math> has positive solutions <math>a</math> and <math>b</math>. Then the pair <math>(a,b)</math> is<br />
<br />
<math> \mathrm{(A) \ } (-2,1)\qquad \mathrm{(B) \ } (-1,2)\qquad \mathrm{(C) \ } (1,-2)\qquad \mathrm{(D) \ } (2,-1)\qquad \mathrm{(E) \ } (4,4) </math><br />
<br />
[[2002 AMC 10B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
The product of three consecutive positive integers is <math>8</math> times their sum. What is the sum of the squares?<br />
<br />
<math> \mathrm{(A) \ } 50\qquad \mathrm{(B) \ } 77\qquad \mathrm{(C) \ } 110\qquad \mathrm{(D) \ } 149\qquad \mathrm{(E) \ } 194 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
For which of the following values of <math>k</math> does the equation <math>\frac{x-1}{x-2} = \frac{x-k}{x-6}</math> have no solution for <math>x</math>?<br />
<br />
<math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5</math><br />
<br />
[[2002 AMC 10B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13==<br />
<br />
Find the value(s) of <math>x</math> such that <math>8xy - 12y + 2x - 3 = 0</math> is true for all values of <math>y</math>.<br />
<br />
<math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac32 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14</math><br />
<br />
<br />
[[2002 AMC 10B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
The number <math>25^{64}\cdot 64^{25}</math> is the square of a positive integer <math>N</math>. In decimal representation, the sum of the digits of <math>N</math> is<br />
<br />
<math> \mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 35 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The positive integers <math>A</math>, <math>B</math>, <math>A-B</math>, and <math>A+B</math> are all prime numbers. The sum of these four primes is<br />
<br />
<math> \mathrm{(A) \ } \text{even}\qquad \mathrm{(B) \ } \text{divisible by }3\qquad \mathrm{(C) \ } \text{divisible by }5\qquad \mathrm{(D) \ } \text{divisible by }7\qquad \mathrm{(E) \ } \text{prime}</math><br />
<br />
[[2002 AMC 10B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
For how many integers <math>n</math> is <math>\frac{n}{20-n}</math> the square of an integer?<br />
<br />
<math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 10</math><br />
<br />
<br />
[[2002 AMC 10B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
A regular octagon <math>ABCDEFGH</math> has sides of length two. Find the area of <math>\triangle ADG</math>.<br />
<br />
<math>\textbf{(A) } 4 + 2\sqrt2 \qquad \textbf{(B) } 6 + \sqrt2\qquad \textbf{(C) } 4 + 3\sqrt2 \qquad \textbf{(D) } 3 + 4\sqrt2 \qquad \textbf{(E) } 8 + \sqrt2</math><br />
<br />
[[2002 AMC 10B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?<br />
<br />
<math>\textbf{(A) } 8\qquad \textbf{(B) } 9\qquad \textbf{(C) } 10\qquad \textbf{(D) } 12\qquad \textbf{(E) } 16</math><br />
<br />
[[2002 AMC 10B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
Let <math>\triangle{XOY}</math> be a right-triangle with <math>m\angle{XOY}=90^\circ</math>. Let <math>M</math> and <math>N</math> be the midpoints of the legs <math>OX</math> and <math>OY</math>, respectively. Given <math>XN=19</math> and <math>YM=22</math>, find <math>XY</math>.<br />
<br />
<math> \mathrm{(A) \ } 24\qquad \mathrm{(B) \ } 26\qquad \mathrm{(C) \ } 28\qquad \mathrm{(D) \ } 30\qquad \mathrm{(E) \ } 32 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list?<br />
<br />
<math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2002_AMC_10B_Problems/Problem_25&diff=340372002 AMC 10B Problems/Problem 252010-03-26T18:15:51Z<p>YottaByte: </p>
<hr />
<div>== Problem ==<br />
When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list?<br />
<br />
<math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8 </math><br />
<br />
== Solution ==</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2002_AMC_10B_Problems&diff=340362002 AMC 10B Problems2010-03-26T18:13:57Z<p>YottaByte: /* Problem 18 */</p>
<hr />
<div>==Problem 1==<br />
The ratio <math>\frac{2^{2001}\cdot3^{2003}}{6^{2002}}</math> is:<br />
<br />
<math> \mathrm{(A) \ } 1/6\qquad \mathrm{(B) \ } 1/3\qquad \mathrm{(C) \ } 1/2\qquad \mathrm{(D) \ } 2/3\qquad \mathrm{(E) \ } 3/2 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
For the nonzero numbers a, b, and c, define<br />
<br />
<math>(a,b,c)=\frac{abc}{a+b+c}</math><br />
<br />
Find <math>(2,4,6)</math>.<br />
<br />
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 24 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The arithmetic mean of the nine numbers in the set <math>\{9,99,999,9999,\ldots,999999999\}</math> is a 9-digit number <math>M</math>, all of whose digits are distinct. The number <math>M</math> does not contain the digit<br />
<br />
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 8 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
What is the value of<br />
<br />
<math>(3x-2)(4x+1)-(3x-2)4x+1</math><br />
<br />
when <math>x=4</math>?<br />
<br />
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
For how many positive integers n is <math>n^2-3n+2</math> a prime number?<br />
<br />
<math> \mathrm{(A) \ } \text{none}\qquad \mathrm{(B) \ } \text{one}\qquad \mathrm{(C) \ } \text{two}\qquad \mathrm{(D) \ } \text{more than two, but finitely many}\qquad \mathrm{(E) \ } \text{infinitely many} </math><br />
<br />
[[2002 AMC 10B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
Let <math>n</math> be a positive integer such that <math>\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}</math> is an integer. Which of the following statements is '''not''' true?<br />
<br />
<math> \mathrm{(A) \ } 2\text{ divides }n\qquad \mathrm{(B) \ } 3\text{ divides }n\qquad \mathrm{(C) \ } 6\text{ divides }n\qquad \mathrm{(D) \ } 7\text{ divides }n\qquad \mathrm{(E) \ } n>84 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
<br />
Suppose July of year <math>N</math> has five Mondays. Which of the following must occurs five times in the August of year <math>N</math>? (Note: Both months have <math>31</math> days.)<br />
<br />
<math>\textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Thursday} \qquad \textbf{(E)}\ \text{Friday}</math><br />
<br />
[[2002 AMC 10B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
<br />
Using the letters <math>A</math>, <math>M</math>, <math>O</math>, <math>S</math>, and <math>U</math>, we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" <math>USAMO</math> occupies position<br />
<br />
<math> \mathrm{(A) \ } 112\qquad \mathrm{(B) \ } 113\qquad \mathrm{(C) \ } 114\qquad \mathrm{(D) \ } 115\qquad \mathrm{(E) \ } 116 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
Suppose that <math>a</math> and <math>b</math> are nonzero real numbers, and that the equation <math>x^2+ax+b=0</math> has positive solutions <math>a</math> and <math>b</math>. Then the pair <math>(a,b)</math> is<br />
<br />
<math> \mathrm{(A) \ } (-2,1)\qquad \mathrm{(B) \ } (-1,2)\qquad \mathrm{(C) \ } (1,-2)\qquad \mathrm{(D) \ } (2,-1)\qquad \mathrm{(E) \ } (4,4) </math><br />
<br />
[[2002 AMC 10B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
The product of three consecutive positive integers is <math>8</math> times their sum. What is the sum of the squares?<br />
<br />
<math> \mathrm{(A) \ } 50\qquad \mathrm{(B) \ } 77\qquad \mathrm{(C) \ } 110\qquad \mathrm{(D) \ } 149\qquad \mathrm{(E) \ } 194 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
For which of the following values of <math>k</math> does the equation <math>\frac{x-1}{x-2} = \frac{x-k}{x-6}</math> have no solution for <math>x</math>?<br />
<br />
<math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5</math><br />
<br />
[[2002 AMC 10B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13==<br />
<br />
Find the value(s) of <math>x</math> such that <math>8xy - 12y + 2x - 3 = 0</math> is true for all values of <math>y</math>.<br />
<br />
<math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac32 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14</math><br />
<br />
<br />
[[2002 AMC 10B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
The number <math>25^{64}\cdot 64^{25}</math> is the square of a positive integer <math>N</math>. In decimal representation, the sum of the digits of <math>N</math> is<br />
<br />
<math> \mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 35 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The positive integers <math>A</math>, <math>B</math>, <math>A-B</math>, and <math>A+B</math> are all prime numbers. The sum of these four primes is<br />
<br />
<math> \mathrm{(A) \ } \text{even}\qquad \mathrm{(B) \ } \text{divisible by }3\qquad \mathrm{(C) \ } \text{divisible by }5\qquad \mathrm{(D) \ } \text{divisible by }7\qquad \mathrm{(E) \ } \text{prime}</math><br />
<br />
[[2002 AMC 10B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
For how many integers <math>n</math> is <math>\frac{n}{20-n}</math> the square of an integer?<br />
<br />
<math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 10</math><br />
<br />
<br />
[[2002 AMC 10B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
A regular octagon <math>ABCDEFGH</math> has sides of length two. Find the area of <math>\triangle ADG</math>.<br />
<br />
<math>\textbf{(A) } 4 + 2\sqrt2 \qquad \textbf{(B) } 6 + \sqrt2\qquad \textbf{(C) } 4 + 3\sqrt2 \qquad \textbf{(D) } 3 + 4\sqrt2 \qquad \textbf{(E) } 8 + \sqrt2</math><br />
<br />
[[2002 AMC 10B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?<br />
<br />
<math>\textbf{(A) } 8\qquad \textbf{(B) } 9\qquad \textbf{(C) } 10\qquad \textbf{(D) } 12\qquad \textbf{(E) } 16</math><br />
<br />
[[2002 AMC 10B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
Let <math>\triangle{XOY}</math> be a right-triangle with <math>m\angle{XOY}=90^\circ</math>. Let <math>M</math> and <math>N</math> be the midpoints of the legs <math>OX</math> and <math>OY</math>, respectively. Given <math>XN=19</math> and <math>YM=22</math>, find <math>XY</math>.<br />
<br />
<math> \mathrm{(A) \ } 24\qquad \mathrm{(B) \ } 26\qquad \mathrm{(C) \ } 28\qquad \mathrm{(D) \ } 30\qquad \mathrm{(E) \ } 32 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2002_AMC_10B_Problems&diff=340352002 AMC 10B Problems2010-03-26T18:13:18Z<p>YottaByte: /* Problem 18 */</p>
<hr />
<div>==Problem 1==<br />
The ratio <math>\frac{2^{2001}\cdot3^{2003}}{6^{2002}}</math> is:<br />
<br />
<math> \mathrm{(A) \ } 1/6\qquad \mathrm{(B) \ } 1/3\qquad \mathrm{(C) \ } 1/2\qquad \mathrm{(D) \ } 2/3\qquad \mathrm{(E) \ } 3/2 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
For the nonzero numbers a, b, and c, define<br />
<br />
<math>(a,b,c)=\frac{abc}{a+b+c}</math><br />
<br />
Find <math>(2,4,6)</math>.<br />
<br />
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 24 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The arithmetic mean of the nine numbers in the set <math>\{9,99,999,9999,\ldots,999999999\}</math> is a 9-digit number <math>M</math>, all of whose digits are distinct. The number <math>M</math> does not contain the digit<br />
<br />
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 8 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
What is the value of<br />
<br />
<math>(3x-2)(4x+1)-(3x-2)4x+1</math><br />
<br />
when <math>x=4</math>?<br />
<br />
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
For how many positive integers n is <math>n^2-3n+2</math> a prime number?<br />
<br />
<math> \mathrm{(A) \ } \text{none}\qquad \mathrm{(B) \ } \text{one}\qquad \mathrm{(C) \ } \text{two}\qquad \mathrm{(D) \ } \text{more than two, but finitely many}\qquad \mathrm{(E) \ } \text{infinitely many} </math><br />
<br />
[[2002 AMC 10B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
Let <math>n</math> be a positive integer such that <math>\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}</math> is an integer. Which of the following statements is '''not''' true?<br />
<br />
<math> \mathrm{(A) \ } 2\text{ divides }n\qquad \mathrm{(B) \ } 3\text{ divides }n\qquad \mathrm{(C) \ } 6\text{ divides }n\qquad \mathrm{(D) \ } 7\text{ divides }n\qquad \mathrm{(E) \ } n>84 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
<br />
Suppose July of year <math>N</math> has five Mondays. Which of the following must occurs five times in the August of year <math>N</math>? (Note: Both months have <math>31</math> days.)<br />
<br />
<math>\textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Thursday} \qquad \textbf{(E)}\ \text{Friday}</math><br />
<br />
[[2002 AMC 10B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
<br />
Using the letters <math>A</math>, <math>M</math>, <math>O</math>, <math>S</math>, and <math>U</math>, we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" <math>USAMO</math> occupies position<br />
<br />
<math> \mathrm{(A) \ } 112\qquad \mathrm{(B) \ } 113\qquad \mathrm{(C) \ } 114\qquad \mathrm{(D) \ } 115\qquad \mathrm{(E) \ } 116 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
Suppose that <math>a</math> and <math>b</math> are nonzero real numbers, and that the equation <math>x^2+ax+b=0</math> has positive solutions <math>a</math> and <math>b</math>. Then the pair <math>(a,b)</math> is<br />
<br />
<math> \mathrm{(A) \ } (-2,1)\qquad \mathrm{(B) \ } (-1,2)\qquad \mathrm{(C) \ } (1,-2)\qquad \mathrm{(D) \ } (2,-1)\qquad \mathrm{(E) \ } (4,4) </math><br />
<br />
[[2002 AMC 10B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
The product of three consecutive positive integers is <math>8</math> times their sum. What is the sum of the squares?<br />
<br />
<math> \mathrm{(A) \ } 50\qquad \mathrm{(B) \ } 77\qquad \mathrm{(C) \ } 110\qquad \mathrm{(D) \ } 149\qquad \mathrm{(E) \ } 194 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
For which of the following values of <math>k</math> does the equation <math>\frac{x-1}{x-2} = \frac{x-k}{x-6}</math> have no solution for <math>x</math>?<br />
<br />
<math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5</math><br />
<br />
[[2002 AMC 10B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13==<br />
<br />
Find the value(s) of <math>x</math> such that <math>8xy - 12y + 2x - 3 = 0</math> is true for all values of <math>y</math>.<br />
<br />
<math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac32 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14</math><br />
<br />
<br />
[[2002 AMC 10B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
The number <math>25^{64}\cdot 64^{25}</math> is the square of a positive integer <math>N</math>. In decimal representation, the sum of the digits of <math>N</math> is<br />
<br />
<math> \mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 35 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The positive integers <math>A</math>, <math>B</math>, <math>A-B</math>, and <math>A+B</math> are all prime numbers. The sum of these four primes is<br />
<br />
<math> \mathrm{(A) \ } \text{even}\qquad \mathrm{(B) \ } \text{divisible by }3\qquad \mathrm{(C) \ } \text{divisible by }5\qquad \mathrm{(D) \ } \text{divisible by }7\qquad \mathrm{(E) \ } \text{prime}</math><br />
<br />
[[2002 AMC 10B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
For how many integers <math>n</math> is <math>\frac{n}{20-n}</math> the square of an integer?<br />
<br />
<math>\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 10</math><br />
<br />
<br />
[[2002 AMC 10B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
A regular octagon <math>ABCDEFGH</math> has sides of length two. Find the area of <math>\triangle ADG</math>.<br />
<br />
<math>\textbf{(A) } 4 + 2\sqrt2 \qquad \textbf{(B) } 6 + \sqrt2\qquad \textbf{(C) } 4 + 3\sqrt2 \qquad \textbf{(D) } 3 + 4\sqrt2 \qquad \textbf{(E) } 8 + \sqrt2</math><br />
<br />
[[2002 AMC 10B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?<br />
<br />
<math>\textbf{(A) } 8\qquad \textbf{(B) } 9\qquad \textbf{(C) } 10\qquad \textbf{(D) } 12\qquad \textbf{(E) } 16\</math><br />
<br />
[[2002 AMC 10B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
Let <math>\triangle{XOY}</math> be a right-triangle with <math>m\angle{XOY}=90^\circ</math>. Let <math>M</math> and <math>N</math> be the midpoints of the legs <math>OX</math> and <math>OY</math>, respectively. Given <math>XN=19</math> and <math>YM=22</math>, find <math>XY</math>.<br />
<br />
<math> \mathrm{(A) \ } 24\qquad \mathrm{(B) \ } 26\qquad \mathrm{(C) \ } 28\qquad \mathrm{(D) \ } 30\qquad \mathrm{(E) \ } 32 </math><br />
<br />
[[2002 AMC 10B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2002 AMC 10B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC Problems and Solutions]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2002_AMC_10B_Problems/Problem_18&diff=340342002 AMC 10B Problems/Problem 182010-03-26T18:12:53Z<p>YottaByte: /* Problem */</p>
<hr />
<div>== Problem ==<br />
<br />
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?<br />
<br />
<math>\textbf{(A) } 8\qquad \textbf{(B) } 9\qquad \textbf{(C) } 10\qquad \textbf{(D) } 12\qquad \textbf{(E) } 16\</math><br />
<br />
== Solution ==</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=340332005 AMC 12B Problems2010-03-26T16:34:27Z<p>YottaByte: /* Problem 25 */</p>
<hr />
<div>{{empty}}<br />
== Problem 1 ==<br />
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars?<br />
<br />
<math><br />
\mathrm{(A)}\ 100 \qquad<br />
\mathrm{(B)}\ 200 \qquad<br />
\mathrm{(C)}\ 300 \qquad<br />
\mathrm{(D)}\ 400 \qquad<br />
\mathrm{(E)}\ 500<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>?<br />
<math><br />
\mathrm{(A)}\ 2 \qquad<br />
\mathrm{(B)}\ 4 \qquad<br />
\mathrm{(C)}\ 10 \qquad<br />
\mathrm{(D)}\ 20 \qquad<br />
\mathrm{(E)}\ 40<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br />
<br />
<math><br />
\mathrm{(A)}\ \frac15 \qquad<br />
\mathrm{(B)}\ \frac13 \qquad<br />
\mathrm{(C)}\ \frac25 \qquad<br />
\mathrm{(D)}\ \frac23 \qquad<br />
\mathrm{(E)}\ \frac45<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> quizzes for the year. She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br />
<br />
<asy><br />
unitsize(2cm);<br />
defaultpen(linewidth(.8pt));<br />
fill(unitsquare,gray);<br />
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br />
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br />
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br />
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br />
</asy><br />
<br />
<math><br />
\mathrm{(A)}\ 80-20\pi \qquad<br />
\mathrm{(B)}\ 60-10\pi \qquad<br />
\mathrm{(C)}\ 80-10\pi \qquad<br />
\mathrm{(D)}\ 60+10\pi \qquad<br />
\mathrm{(E)}\ 80+10\pi<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
In <math>\triangle ABC</math>, we have <math>AC=BC=7</math> and <math>AB=2</math>. Suppose that <math>D</math> is a point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 3 \qquad<br />
\mathrm{(B)}\ 2\sqrt{3} \qquad<br />
\mathrm{(C)}\ 4 \qquad<br />
\mathrm{(D)}\ 5 \qquad<br />
\mathrm{(E)}\ 4\sqrt{2}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 6 \qquad<br />
\mathrm{(B)}\ 12 \qquad<br />
\mathrm{(C)}\ 16 \qquad<br />
\mathrm{(D)}\ 24 \qquad<br />
\mathrm{(E)}\ 25<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the<br />
vertex of the parabola <math>y = x^2 + a^2</math> ?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 2 \qquad<br />
\mathrm{(D)}\ 10 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
[[2005 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?<br />
<br />
<math>\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
The [[quadratic equation]] <math>x^2+mx+n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>?<br />
<br />
<math>\mathrm{(A)}\ {{{1}}} \qquad \mathrm{(B)}\ {{{2}}} \qquad \mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?<br />
<br />
<math><br />
\mathrm (A)\ \sqrt{2} \qquad<br />
\mathrm (B)\ \sqrt{3} \qquad<br />
\mathrm (C)\ 1+\sqrt{2}\qquad<br />
\mathrm (D)\ 1+\sqrt{3}\qquad<br />
\mathrm (E)\ 3<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
How many distinct four-tuples <math>(a, b, c, d)</math> of rational numbers are there with<br />
<br />
<math>a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 17 \qquad<br />
\mathrm{(D)}\ 2004 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R</math> as the region in the first quadrant consisting of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math>R</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 25 \qquad<br />
\mathrm{(B)}\ 39 \qquad<br />
\mathrm{(C)}\ 51 \qquad<br />
\mathrm{(D)}\ 60 \qquad<br />
\mathrm{(E)}\ 80 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^{2}-y^{2}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 88 \qquad<br />
\mathrm{(B)}\ 112 \qquad<br />
\mathrm{(C)}\ 116 \qquad<br />
\mathrm{(D)}\ 144 \qquad<br />
\mathrm{(E)}\ 154 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set<br />
<br />
<cmath>\{-7,-5,-3,-2,2,4,6,13\}.</cmath><br />
<br />
What is the minimum possible value of<br />
<br />
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath><br />
<br />
<math><br />
\mathrm{(A)}\ 30 \qquad<br />
\mathrm{(B)}\ 32 \qquad<br />
\mathrm{(C)}\ 34 \qquad<br />
\mathrm{(D)}\ 40 \qquad<br />
\mathrm{(E)}\ 50<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule<br />
<br />
<cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath><br />
<br />
where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</math>. How many possible values are there for <math>z_{0}</math>?<br />
<br />
<math><br />
\textbf{(A)}\ 1 \qquad <br />
\textbf{(B)}\ 2 \qquad <br />
\textbf{(C)}\ 4 \qquad <br />
\textbf{(D)}\ 2005 \qquad <br />
\textbf{(E)}\ 2^{2005}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which<br />
<br />
<cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath><br />
There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y.z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math><br />
<br />
<math><br />
\textbf{(A)}\ \frac {15}{2} \qquad <br />
\textbf{(B)}\ \frac {29}{2} \qquad <br />
\textbf{(C)}\ 15 \qquad <br />
\textbf{(D)}\ \frac {39}{2} \qquad <br />
\textbf{(E)}\ 24<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
Six ants simultaneously stand on the six [[vertex|vertices]] of a regular [[octahedron]], with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex?<br />
<br />
<math>\mathrm{(A)}\ \frac {5}{256}<br />
\qquad\mathrm{(B)}\ \frac {21}{1024}<br />
\qquad\mathrm{(C)}\ \frac {11}{512}<br />
\qquad\mathrm{(D)}\ \frac {23}{1024}<br />
\qquad\mathrm{(E)}\ \frac {3}{128}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC 12]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[2005 AMC 12B]]<br />
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=340322005 AMC 12B Problems2010-03-26T16:33:27Z<p>YottaByte: /* Problem 23 */</p>
<hr />
<div>{{empty}}<br />
== Problem 1 ==<br />
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars?<br />
<br />
<math><br />
\mathrm{(A)}\ 100 \qquad<br />
\mathrm{(B)}\ 200 \qquad<br />
\mathrm{(C)}\ 300 \qquad<br />
\mathrm{(D)}\ 400 \qquad<br />
\mathrm{(E)}\ 500<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>?<br />
<math><br />
\mathrm{(A)}\ 2 \qquad<br />
\mathrm{(B)}\ 4 \qquad<br />
\mathrm{(C)}\ 10 \qquad<br />
\mathrm{(D)}\ 20 \qquad<br />
\mathrm{(E)}\ 40<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br />
<br />
<math><br />
\mathrm{(A)}\ \frac15 \qquad<br />
\mathrm{(B)}\ \frac13 \qquad<br />
\mathrm{(C)}\ \frac25 \qquad<br />
\mathrm{(D)}\ \frac23 \qquad<br />
\mathrm{(E)}\ \frac45<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> quizzes for the year. She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br />
<br />
<asy><br />
unitsize(2cm);<br />
defaultpen(linewidth(.8pt));<br />
fill(unitsquare,gray);<br />
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br />
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br />
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br />
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br />
</asy><br />
<br />
<math><br />
\mathrm{(A)}\ 80-20\pi \qquad<br />
\mathrm{(B)}\ 60-10\pi \qquad<br />
\mathrm{(C)}\ 80-10\pi \qquad<br />
\mathrm{(D)}\ 60+10\pi \qquad<br />
\mathrm{(E)}\ 80+10\pi<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
In <math>\triangle ABC</math>, we have <math>AC=BC=7</math> and <math>AB=2</math>. Suppose that <math>D</math> is a point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 3 \qquad<br />
\mathrm{(B)}\ 2\sqrt{3} \qquad<br />
\mathrm{(C)}\ 4 \qquad<br />
\mathrm{(D)}\ 5 \qquad<br />
\mathrm{(E)}\ 4\sqrt{2}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 6 \qquad<br />
\mathrm{(B)}\ 12 \qquad<br />
\mathrm{(C)}\ 16 \qquad<br />
\mathrm{(D)}\ 24 \qquad<br />
\mathrm{(E)}\ 25<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the<br />
vertex of the parabola <math>y = x^2 + a^2</math> ?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 2 \qquad<br />
\mathrm{(D)}\ 10 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
[[2005 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?<br />
<br />
<math>\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
The [[quadratic equation]] <math>x^2+mx+n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>?<br />
<br />
<math>\mathrm{(A)}\ {{{1}}} \qquad \mathrm{(B)}\ {{{2}}} \qquad \mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?<br />
<br />
<math><br />
\mathrm (A)\ \sqrt{2} \qquad<br />
\mathrm (B)\ \sqrt{3} \qquad<br />
\mathrm (C)\ 1+\sqrt{2}\qquad<br />
\mathrm (D)\ 1+\sqrt{3}\qquad<br />
\mathrm (E)\ 3<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
How many distinct four-tuples <math>(a, b, c, d)</math> of rational numbers are there with<br />
<br />
<math>a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 17 \qquad<br />
\mathrm{(D)}\ 2004 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R</math> as the region in the first quadrant consisting of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math>R</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 25 \qquad<br />
\mathrm{(B)}\ 39 \qquad<br />
\mathrm{(C)}\ 51 \qquad<br />
\mathrm{(D)}\ 60 \qquad<br />
\mathrm{(E)}\ 80 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^{2}-y^{2}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 88 \qquad<br />
\mathrm{(B)}\ 112 \qquad<br />
\mathrm{(C)}\ 116 \qquad<br />
\mathrm{(D)}\ 144 \qquad<br />
\mathrm{(E)}\ 154 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set<br />
<br />
<cmath>\{-7,-5,-3,-2,2,4,6,13\}.</cmath><br />
<br />
What is the minimum possible value of<br />
<br />
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath><br />
<br />
<math><br />
\mathrm{(A)}\ 30 \qquad<br />
\mathrm{(B)}\ 32 \qquad<br />
\mathrm{(C)}\ 34 \qquad<br />
\mathrm{(D)}\ 40 \qquad<br />
\mathrm{(E)}\ 50<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule<br />
<br />
<cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath><br />
<br />
where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</math>. How many possible values are there for <math>z_{0}</math>?<br />
<br />
<math><br />
\textbf{(A)}\ 1 \qquad <br />
\textbf{(B)}\ 2 \qquad <br />
\textbf{(C)}\ 4 \qquad <br />
\textbf{(D)}\ 2005 \qquad <br />
\textbf{(E)}\ 2^{2005}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which<br />
<br />
<cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath><br />
There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y.z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math><br />
<br />
<math><br />
\textbf{(A)}\ \frac {15}{2} \qquad <br />
\textbf{(B)}\ \frac {29}{2} \qquad <br />
\textbf{(C)}\ 15 \qquad <br />
\textbf{(D)}\ \frac {39}{2} \qquad <br />
\textbf{(E)}\ 24<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC 12]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[2005 AMC 12B]]<br />
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=340312005 AMC 12B Problems2010-03-26T16:32:57Z<p>YottaByte: /* Problem 22 */</p>
<hr />
<div>{{empty}}<br />
== Problem 1 ==<br />
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars?<br />
<br />
<math><br />
\mathrm{(A)}\ 100 \qquad<br />
\mathrm{(B)}\ 200 \qquad<br />
\mathrm{(C)}\ 300 \qquad<br />
\mathrm{(D)}\ 400 \qquad<br />
\mathrm{(E)}\ 500<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>?<br />
<math><br />
\mathrm{(A)}\ 2 \qquad<br />
\mathrm{(B)}\ 4 \qquad<br />
\mathrm{(C)}\ 10 \qquad<br />
\mathrm{(D)}\ 20 \qquad<br />
\mathrm{(E)}\ 40<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br />
<br />
<math><br />
\mathrm{(A)}\ \frac15 \qquad<br />
\mathrm{(B)}\ \frac13 \qquad<br />
\mathrm{(C)}\ \frac25 \qquad<br />
\mathrm{(D)}\ \frac23 \qquad<br />
\mathrm{(E)}\ \frac45<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> quizzes for the year. She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br />
<br />
<asy><br />
unitsize(2cm);<br />
defaultpen(linewidth(.8pt));<br />
fill(unitsquare,gray);<br />
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br />
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br />
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br />
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br />
</asy><br />
<br />
<math><br />
\mathrm{(A)}\ 80-20\pi \qquad<br />
\mathrm{(B)}\ 60-10\pi \qquad<br />
\mathrm{(C)}\ 80-10\pi \qquad<br />
\mathrm{(D)}\ 60+10\pi \qquad<br />
\mathrm{(E)}\ 80+10\pi<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
In <math>\triangle ABC</math>, we have <math>AC=BC=7</math> and <math>AB=2</math>. Suppose that <math>D</math> is a point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 3 \qquad<br />
\mathrm{(B)}\ 2\sqrt{3} \qquad<br />
\mathrm{(C)}\ 4 \qquad<br />
\mathrm{(D)}\ 5 \qquad<br />
\mathrm{(E)}\ 4\sqrt{2}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 6 \qquad<br />
\mathrm{(B)}\ 12 \qquad<br />
\mathrm{(C)}\ 16 \qquad<br />
\mathrm{(D)}\ 24 \qquad<br />
\mathrm{(E)}\ 25<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the<br />
vertex of the parabola <math>y = x^2 + a^2</math> ?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 2 \qquad<br />
\mathrm{(D)}\ 10 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
[[2005 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?<br />
<br />
<math>\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
The [[quadratic equation]] <math>x^2+mx+n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>?<br />
<br />
<math>\mathrm{(A)}\ {{{1}}} \qquad \mathrm{(B)}\ {{{2}}} \qquad \mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?<br />
<br />
<math><br />
\mathrm (A)\ \sqrt{2} \qquad<br />
\mathrm (B)\ \sqrt{3} \qquad<br />
\mathrm (C)\ 1+\sqrt{2}\qquad<br />
\mathrm (D)\ 1+\sqrt{3}\qquad<br />
\mathrm (E)\ 3<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
How many distinct four-tuples <math>(a, b, c, d)</math> of rational numbers are there with<br />
<br />
<math>a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 17 \qquad<br />
\mathrm{(D)}\ 2004 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R</math> as the region in the first quadrant consisting of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math>R</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 25 \qquad<br />
\mathrm{(B)}\ 39 \qquad<br />
\mathrm{(C)}\ 51 \qquad<br />
\mathrm{(D)}\ 60 \qquad<br />
\mathrm{(E)}\ 80 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^{2}-y^{2}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 88 \qquad<br />
\mathrm{(B)}\ 112 \qquad<br />
\mathrm{(C)}\ 116 \qquad<br />
\mathrm{(D)}\ 144 \qquad<br />
\mathrm{(E)}\ 154 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set<br />
<br />
<cmath>\{-7,-5,-3,-2,2,4,6,13\}.</cmath><br />
<br />
What is the minimum possible value of<br />
<br />
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath><br />
<br />
<math><br />
\mathrm{(A)}\ 30 \qquad<br />
\mathrm{(B)}\ 32 \qquad<br />
\mathrm{(C)}\ 34 \qquad<br />
\mathrm{(D)}\ 40 \qquad<br />
\mathrm{(E)}\ 50<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule<br />
<br />
<cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath><br />
<br />
where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</math>. How many possible values are there for <math>z_{0}</math>?<br />
<br />
<math><br />
\textbf{(A)}\ 1 \qquad <br />
\textbf{(B)}\ 2 \qquad <br />
\textbf{(C)}\ 4 \qquad <br />
\textbf{(D)}\ 2005 \qquad <br />
\textbf{(E)}\ 2^{2005}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC 12]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[2005 AMC 12B]]<br />
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems/Problem_14&diff=340302005 AMC 12B Problems/Problem 142010-03-26T16:31:55Z<p>YottaByte: </p>
<hr />
<div>{{empty}}<br />
== Problem ==<br />
{{problem}}<br />
== Solution ==<br />
<br />
== See also ==<br />
* [[2005 AMC 12B Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems/Problem_15&diff=340292005 AMC 12B Problems/Problem 152010-03-26T16:31:18Z<p>YottaByte: /* Problem */</p>
<hr />
<div>== Problem ==<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
* [[2005 AMC 12B Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems/Problem_24&diff=340282005 AMC 12B Problems/Problem 242010-03-26T16:30:34Z<p>YottaByte: </p>
<hr />
<div>{{empty}}<br />
== Problem ==<br />
{{problem}}<br />
== Solution ==<br />
<br />
== See also ==<br />
* [[2005 AMC 12B Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems/Problem_21&diff=340272005 AMC 12B Problems/Problem 212010-03-26T16:30:01Z<p>YottaByte: </p>
<hr />
<div>{{empty}}<br />
== Problem ==<br />
{{problem}}<br />
== Solution ==<br />
<br />
== See also ==<br />
* [[2005 AMC 12B Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems/Problem_13&diff=340262005 AMC 12B Problems/Problem 132010-03-26T16:29:17Z<p>YottaByte: </p>
<hr />
<div>{{empty}}<br />
== Problem ==<br />
{{problem}}<br />
== Solution ==<br />
<br />
== See also ==<br />
* [[2005 AMC 12B Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems/Problem_11&diff=340252005 AMC 12B Problems/Problem 112010-03-26T16:28:04Z<p>YottaByte: </p>
<hr />
<div>{{empty}}<br />
== Problem ==<br />
{{problem}}<br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
* [[2005 AMC 12B Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems/Problem_11&diff=340242005 AMC 12B Problems/Problem 112010-03-26T16:27:02Z<p>YottaByte: /* Problem */</p>
<hr />
<div>== Problem ==<br />
{{problem}}<br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
* [[2005 AMC 12B Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=340232005 AMC 12B Problems2010-03-26T16:25:49Z<p>YottaByte: /* Problem 16 */</p>
<hr />
<div>{{empty}}<br />
== Problem 1 ==<br />
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars?<br />
<br />
<math><br />
\mathrm{(A)}\ 100 \qquad<br />
\mathrm{(B)}\ 200 \qquad<br />
\mathrm{(C)}\ 300 \qquad<br />
\mathrm{(D)}\ 400 \qquad<br />
\mathrm{(E)}\ 500<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>?<br />
<math><br />
\mathrm{(A)}\ 2 \qquad<br />
\mathrm{(B)}\ 4 \qquad<br />
\mathrm{(C)}\ 10 \qquad<br />
\mathrm{(D)}\ 20 \qquad<br />
\mathrm{(E)}\ 40<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br />
<br />
<math><br />
\mathrm{(A)}\ \frac15 \qquad<br />
\mathrm{(B)}\ \frac13 \qquad<br />
\mathrm{(C)}\ \frac25 \qquad<br />
\mathrm{(D)}\ \frac23 \qquad<br />
\mathrm{(E)}\ \frac45<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> quizzes for the year. She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br />
<br />
<asy><br />
unitsize(2cm);<br />
defaultpen(linewidth(.8pt));<br />
fill(unitsquare,gray);<br />
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br />
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br />
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br />
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br />
</asy><br />
<br />
<math><br />
\mathrm{(A)}\ 80-20\pi \qquad<br />
\mathrm{(B)}\ 60-10\pi \qquad<br />
\mathrm{(C)}\ 80-10\pi \qquad<br />
\mathrm{(D)}\ 60+10\pi \qquad<br />
\mathrm{(E)}\ 80+10\pi<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
In <math>\triangle ABC</math>, we have <math>AC=BC=7</math> and <math>AB=2</math>. Suppose that <math>D</math> is a point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 3 \qquad<br />
\mathrm{(B)}\ 2\sqrt{3} \qquad<br />
\mathrm{(C)}\ 4 \qquad<br />
\mathrm{(D)}\ 5 \qquad<br />
\mathrm{(E)}\ 4\sqrt{2}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 6 \qquad<br />
\mathrm{(B)}\ 12 \qquad<br />
\mathrm{(C)}\ 16 \qquad<br />
\mathrm{(D)}\ 24 \qquad<br />
\mathrm{(E)}\ 25<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the<br />
vertex of the parabola <math>y = x^2 + a^2</math> ?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 2 \qquad<br />
\mathrm{(D)}\ 10 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
[[2005 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?<br />
<br />
<math>\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
The [[quadratic equation]] <math>x^2+mx+n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>?<br />
<br />
<math>\mathrm{(A)}\ {{{1}}} \qquad \mathrm{(B)}\ {{{2}}} \qquad \mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?<br />
<br />
<math><br />
\mathrm (A)\ \sqrt{2} \qquad<br />
\mathrm (B)\ \sqrt{3} \qquad<br />
\mathrm (C)\ 1+\sqrt{2}\qquad<br />
\mathrm (D)\ 1+\sqrt{3}\qquad<br />
\mathrm (E)\ 3<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
How many distinct four-tuples <math>(a, b, c, d)</math> of rational numbers are there with<br />
<br />
<math>a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 17 \qquad<br />
\mathrm{(D)}\ 2004 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R</math> as the region in the first quadrant consisting of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math>R</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 25 \qquad<br />
\mathrm{(B)}\ 39 \qquad<br />
\mathrm{(C)}\ 51 \qquad<br />
\mathrm{(D)}\ 60 \qquad<br />
\mathrm{(E)}\ 80 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^{2}-y^{2}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 88 \qquad<br />
\mathrm{(B)}\ 112 \qquad<br />
\mathrm{(C)}\ 116 \qquad<br />
\mathrm{(D)}\ 144 \qquad<br />
\mathrm{(E)}\ 154 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set<br />
<br />
<cmath>\{-7,-5,-3,-2,2,4,6,13\}.</cmath><br />
<br />
What is the minimum possible value of<br />
<br />
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath><br />
<br />
<math><br />
\mathrm{(A)}\ 30 \qquad<br />
\mathrm{(B)}\ 32 \qquad<br />
\mathrm{(C)}\ 34 \qquad<br />
\mathrm{(D)}\ 40 \qquad<br />
\mathrm{(E)}\ 50<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC 12]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[2005 AMC 12B]]<br />
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=340222005 AMC 12B Problems2010-03-26T16:24:57Z<p>YottaByte: /* Problem 12 */</p>
<hr />
<div>{{empty}}<br />
== Problem 1 ==<br />
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars?<br />
<br />
<math><br />
\mathrm{(A)}\ 100 \qquad<br />
\mathrm{(B)}\ 200 \qquad<br />
\mathrm{(C)}\ 300 \qquad<br />
\mathrm{(D)}\ 400 \qquad<br />
\mathrm{(E)}\ 500<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>?<br />
<math><br />
\mathrm{(A)}\ 2 \qquad<br />
\mathrm{(B)}\ 4 \qquad<br />
\mathrm{(C)}\ 10 \qquad<br />
\mathrm{(D)}\ 20 \qquad<br />
\mathrm{(E)}\ 40<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br />
<br />
<math><br />
\mathrm{(A)}\ \frac15 \qquad<br />
\mathrm{(B)}\ \frac13 \qquad<br />
\mathrm{(C)}\ \frac25 \qquad<br />
\mathrm{(D)}\ \frac23 \qquad<br />
\mathrm{(E)}\ \frac45<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> quizzes for the year. She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br />
<br />
<asy><br />
unitsize(2cm);<br />
defaultpen(linewidth(.8pt));<br />
fill(unitsquare,gray);<br />
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br />
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br />
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br />
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br />
</asy><br />
<br />
<math><br />
\mathrm{(A)}\ 80-20\pi \qquad<br />
\mathrm{(B)}\ 60-10\pi \qquad<br />
\mathrm{(C)}\ 80-10\pi \qquad<br />
\mathrm{(D)}\ 60+10\pi \qquad<br />
\mathrm{(E)}\ 80+10\pi<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
In <math>\triangle ABC</math>, we have <math>AC=BC=7</math> and <math>AB=2</math>. Suppose that <math>D</math> is a point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 3 \qquad<br />
\mathrm{(B)}\ 2\sqrt{3} \qquad<br />
\mathrm{(C)}\ 4 \qquad<br />
\mathrm{(D)}\ 5 \qquad<br />
\mathrm{(E)}\ 4\sqrt{2}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 6 \qquad<br />
\mathrm{(B)}\ 12 \qquad<br />
\mathrm{(C)}\ 16 \qquad<br />
\mathrm{(D)}\ 24 \qquad<br />
\mathrm{(E)}\ 25<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the<br />
vertex of the parabola <math>y = x^2 + a^2</math> ?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 2 \qquad<br />
\mathrm{(D)}\ 10 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
[[2005 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?<br />
<br />
<math>\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
The [[quadratic equation]] <math>x^2+mx+n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>?<br />
<br />
<math>\mathrm{(A)}\ {{{1}}} \qquad \mathrm{(B)}\ {{{2}}} \qquad \mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
How many distinct four-tuples <math>(a, b, c, d)</math> of rational numbers are there with<br />
<br />
<math>a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 17 \qquad<br />
\mathrm{(D)}\ 2004 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R</math> as the region in the first quadrant consisting of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math>R</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 25 \qquad<br />
\mathrm{(B)}\ 39 \qquad<br />
\mathrm{(C)}\ 51 \qquad<br />
\mathrm{(D)}\ 60 \qquad<br />
\mathrm{(E)}\ 80 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^{2}-y^{2}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 88 \qquad<br />
\mathrm{(B)}\ 112 \qquad<br />
\mathrm{(C)}\ 116 \qquad<br />
\mathrm{(D)}\ 144 \qquad<br />
\mathrm{(E)}\ 154 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set<br />
<br />
<cmath>\{-7,-5,-3,-2,2,4,6,13\}.</cmath><br />
<br />
What is the minimum possible value of<br />
<br />
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath><br />
<br />
<math><br />
\mathrm{(A)}\ 30 \qquad<br />
\mathrm{(B)}\ 32 \qquad<br />
\mathrm{(C)}\ 34 \qquad<br />
\mathrm{(D)}\ 40 \qquad<br />
\mathrm{(E)}\ 50<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC 12]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[2005 AMC 12B]]<br />
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=340212005 AMC 12B Problems2010-03-26T16:23:56Z<p>YottaByte: /* Problem 10 */</p>
<hr />
<div>{{empty}}<br />
== Problem 1 ==<br />
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars?<br />
<br />
<math><br />
\mathrm{(A)}\ 100 \qquad<br />
\mathrm{(B)}\ 200 \qquad<br />
\mathrm{(C)}\ 300 \qquad<br />
\mathrm{(D)}\ 400 \qquad<br />
\mathrm{(E)}\ 500<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>?<br />
<math><br />
\mathrm{(A)}\ 2 \qquad<br />
\mathrm{(B)}\ 4 \qquad<br />
\mathrm{(C)}\ 10 \qquad<br />
\mathrm{(D)}\ 20 \qquad<br />
\mathrm{(E)}\ 40<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br />
<br />
<math><br />
\mathrm{(A)}\ \frac15 \qquad<br />
\mathrm{(B)}\ \frac13 \qquad<br />
\mathrm{(C)}\ \frac25 \qquad<br />
\mathrm{(D)}\ \frac23 \qquad<br />
\mathrm{(E)}\ \frac45<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> quizzes for the year. She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br />
<br />
<asy><br />
unitsize(2cm);<br />
defaultpen(linewidth(.8pt));<br />
fill(unitsquare,gray);<br />
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br />
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br />
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br />
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br />
</asy><br />
<br />
<math><br />
\mathrm{(A)}\ 80-20\pi \qquad<br />
\mathrm{(B)}\ 60-10\pi \qquad<br />
\mathrm{(C)}\ 80-10\pi \qquad<br />
\mathrm{(D)}\ 60+10\pi \qquad<br />
\mathrm{(E)}\ 80+10\pi<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
In <math>\triangle ABC</math>, we have <math>AC=BC=7</math> and <math>AB=2</math>. Suppose that <math>D</math> is a point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 3 \qquad<br />
\mathrm{(B)}\ 2\sqrt{3} \qquad<br />
\mathrm{(C)}\ 4 \qquad<br />
\mathrm{(D)}\ 5 \qquad<br />
\mathrm{(E)}\ 4\sqrt{2}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 6 \qquad<br />
\mathrm{(B)}\ 12 \qquad<br />
\mathrm{(C)}\ 16 \qquad<br />
\mathrm{(D)}\ 24 \qquad<br />
\mathrm{(E)}\ 25<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the<br />
vertex of the parabola <math>y = x^2 + a^2</math> ?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 2 \qquad<br />
\mathrm{(D)}\ 10 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
[[2005 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?<br />
<br />
<math>\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}</math><br />
<br />
[[2005 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
How many distinct four-tuples <math>(a, b, c, d)</math> of rational numbers are there with<br />
<br />
<math>a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 17 \qquad<br />
\mathrm{(D)}\ 2004 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R</math> as the region in the first quadrant consisting of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math>R</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 25 \qquad<br />
\mathrm{(B)}\ 39 \qquad<br />
\mathrm{(C)}\ 51 \qquad<br />
\mathrm{(D)}\ 60 \qquad<br />
\mathrm{(E)}\ 80 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^{2}-y^{2}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 88 \qquad<br />
\mathrm{(B)}\ 112 \qquad<br />
\mathrm{(C)}\ 116 \qquad<br />
\mathrm{(D)}\ 144 \qquad<br />
\mathrm{(E)}\ 154 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set<br />
<br />
<cmath>\{-7,-5,-3,-2,2,4,6,13\}.</cmath><br />
<br />
What is the minimum possible value of<br />
<br />
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath><br />
<br />
<math><br />
\mathrm{(A)}\ 30 \qquad<br />
\mathrm{(B)}\ 32 \qquad<br />
\mathrm{(C)}\ 34 \qquad<br />
\mathrm{(D)}\ 40 \qquad<br />
\mathrm{(E)}\ 50<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC 12]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[2005 AMC 12B]]<br />
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=340202005 AMC 12B Problems2010-03-26T16:23:23Z<p>YottaByte: /* Problem 9 */</p>
<hr />
<div>{{empty}}<br />
== Problem 1 ==<br />
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars?<br />
<br />
<math><br />
\mathrm{(A)}\ 100 \qquad<br />
\mathrm{(B)}\ 200 \qquad<br />
\mathrm{(C)}\ 300 \qquad<br />
\mathrm{(D)}\ 400 \qquad<br />
\mathrm{(E)}\ 500<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>?<br />
<math><br />
\mathrm{(A)}\ 2 \qquad<br />
\mathrm{(B)}\ 4 \qquad<br />
\mathrm{(C)}\ 10 \qquad<br />
\mathrm{(D)}\ 20 \qquad<br />
\mathrm{(E)}\ 40<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br />
<br />
<math><br />
\mathrm{(A)}\ \frac15 \qquad<br />
\mathrm{(B)}\ \frac13 \qquad<br />
\mathrm{(C)}\ \frac25 \qquad<br />
\mathrm{(D)}\ \frac23 \qquad<br />
\mathrm{(E)}\ \frac45<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> quizzes for the year. She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br />
<br />
<asy><br />
unitsize(2cm);<br />
defaultpen(linewidth(.8pt));<br />
fill(unitsquare,gray);<br />
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br />
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br />
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br />
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br />
</asy><br />
<br />
<math><br />
\mathrm{(A)}\ 80-20\pi \qquad<br />
\mathrm{(B)}\ 60-10\pi \qquad<br />
\mathrm{(C)}\ 80-10\pi \qquad<br />
\mathrm{(D)}\ 60+10\pi \qquad<br />
\mathrm{(E)}\ 80+10\pi<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
In <math>\triangle ABC</math>, we have <math>AC=BC=7</math> and <math>AB=2</math>. Suppose that <math>D</math> is a point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 3 \qquad<br />
\mathrm{(B)}\ 2\sqrt{3} \qquad<br />
\mathrm{(C)}\ 4 \qquad<br />
\mathrm{(D)}\ 5 \qquad<br />
\mathrm{(E)}\ 4\sqrt{2}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 6 \qquad<br />
\mathrm{(B)}\ 12 \qquad<br />
\mathrm{(C)}\ 16 \qquad<br />
\mathrm{(D)}\ 24 \qquad<br />
\mathrm{(E)}\ 25<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the<br />
vertex of the parabola <math>y = x^2 + a^2</math> ?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 2 \qquad<br />
\mathrm{(D)}\ 10 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
[[2005 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
How many distinct four-tuples <math>(a, b, c, d)</math> of rational numbers are there with<br />
<br />
<math>a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 17 \qquad<br />
\mathrm{(D)}\ 2004 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R</math> as the region in the first quadrant consisting of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math>R</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 25 \qquad<br />
\mathrm{(B)}\ 39 \qquad<br />
\mathrm{(C)}\ 51 \qquad<br />
\mathrm{(D)}\ 60 \qquad<br />
\mathrm{(E)}\ 80 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^{2}-y^{2}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 88 \qquad<br />
\mathrm{(B)}\ 112 \qquad<br />
\mathrm{(C)}\ 116 \qquad<br />
\mathrm{(D)}\ 144 \qquad<br />
\mathrm{(E)}\ 154 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set<br />
<br />
<cmath>\{-7,-5,-3,-2,2,4,6,13\}.</cmath><br />
<br />
What is the minimum possible value of<br />
<br />
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath><br />
<br />
<math><br />
\mathrm{(A)}\ 30 \qquad<br />
\mathrm{(B)}\ 32 \qquad<br />
\mathrm{(C)}\ 34 \qquad<br />
\mathrm{(D)}\ 40 \qquad<br />
\mathrm{(E)}\ 50<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC 12]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[2005 AMC 12B]]<br />
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=340192005 AMC 12B Problems2010-03-26T16:23:01Z<p>YottaByte: /* Problem 9 */</p>
<hr />
<div>{{empty}}<br />
== Problem 1 ==<br />
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars?<br />
<br />
<math><br />
\mathrm{(A)}\ 100 \qquad<br />
\mathrm{(B)}\ 200 \qquad<br />
\mathrm{(C)}\ 300 \qquad<br />
\mathrm{(D)}\ 400 \qquad<br />
\mathrm{(E)}\ 500<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>?<br />
<math><br />
\mathrm{(A)}\ 2 \qquad<br />
\mathrm{(B)}\ 4 \qquad<br />
\mathrm{(C)}\ 10 \qquad<br />
\mathrm{(D)}\ 20 \qquad<br />
\mathrm{(E)}\ 40<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br />
<br />
<math><br />
\mathrm{(A)}\ \frac15 \qquad<br />
\mathrm{(B)}\ \frac13 \qquad<br />
\mathrm{(C)}\ \frac25 \qquad<br />
\mathrm{(D)}\ \frac23 \qquad<br />
\mathrm{(E)}\ \frac45<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> quizzes for the year. She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br />
<br />
<asy><br />
unitsize(2cm);<br />
defaultpen(linewidth(.8pt));<br />
fill(unitsquare,gray);<br />
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br />
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br />
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br />
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br />
</asy><br />
<br />
<math><br />
\mathrm{(A)}\ 80-20\pi \qquad<br />
\mathrm{(B)}\ 60-10\pi \qquad<br />
\mathrm{(C)}\ 80-10\pi \qquad<br />
\mathrm{(D)}\ 60+10\pi \qquad<br />
\mathrm{(E)}\ 80+10\pi<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
In <math>\triangle ABC</math>, we have <math>AC=BC=7</math> and <math>AB=2</math>. Suppose that <math>D</math> is a point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 3 \qquad<br />
\mathrm{(B)}\ 2\sqrt{3} \qquad<br />
\mathrm{(C)}\ 4 \qquad<br />
\mathrm{(D)}\ 5 \qquad<br />
\mathrm{(E)}\ 4\sqrt{2}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 6 \qquad<br />
\mathrm{(B)}\ 12 \qquad<br />
\mathrm{(C)}\ 16 \qquad<br />
\mathrm{(D)}\ 24 \qquad<br />
\mathrm{(E)}\ 25<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the<br />
vertex of the parabola <math>y = x^2 + a^2</math> ?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 2 \qquad<br />
\mathrm{(D)}\ 10 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
This is an empty template page which needs to be filled. You can help us out by finding the needed content and editing it in. Thanks.<br />
[[2005 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
The sum of four two-digit numbers is <math>221</math>. Non of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br />
<br />
<math><br />
\mathrm{(A)}\ 1 \qquad<br />
\mathrm{(B)}\ 2 \qquad<br />
\mathrm{(C)}\ 3 \qquad<br />
\mathrm{(D)}\ 4 \qquad<br />
\mathrm{(E)}\ 5<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
How many distinct four-tuples <math>(a, b, c, d)</math> of rational numbers are there with<br />
<br />
<math>a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 0 \qquad<br />
\mathrm{(B)}\ 1 \qquad<br />
\mathrm{(C)}\ 17 \qquad<br />
\mathrm{(D)}\ 2004 \qquad<br />
\mathrm{(E)}\ \text{infinitely many}<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R</math> as the region in the first quadrant consisting of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math>R</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 25 \qquad<br />
\mathrm{(B)}\ 39 \qquad<br />
\mathrm{(C)}\ 51 \qquad<br />
\mathrm{(D)}\ 60 \qquad<br />
\mathrm{(E)}\ 80 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Let <math>x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits of <math>x</math>. The integers <math>x</math> and <math>y</math> satisfy <math>x^{2}-y^{2}=m^{2}</math> for some positive integer <math>m</math>. What is <math>x+y+m</math>?<br />
<br />
<math><br />
\mathrm{(A)}\ 88 \qquad<br />
\mathrm{(B)}\ 112 \qquad<br />
\mathrm{(C)}\ 116 \qquad<br />
\mathrm{(D)}\ 144 \qquad<br />
\mathrm{(E)}\ 154 \qquad<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set<br />
<br />
<cmath>\{-7,-5,-3,-2,2,4,6,13\}.</cmath><br />
<br />
What is the minimum possible value of<br />
<br />
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath><br />
<br />
<math><br />
\mathrm{(A)}\ 30 \qquad<br />
\mathrm{(B)}\ 32 \qquad<br />
\mathrm{(C)}\ 34 \qquad<br />
\mathrm{(D)}\ 40 \qquad<br />
\mathrm{(E)}\ 50<br />
</math><br />
<br />
[[2005 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[2005 AMC 12B Problems/Problem 25|Solution]]<br />
<br />
== See also ==<br />
* [[AMC 12]]<br />
* [[AMC 12 Problems and Solutions]]<br />
* [[2005 AMC 12B]]<br />
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=User:YottaByte&diff=34018User:YottaByte2010-03-26T15:25:02Z<p>YottaByte: </p>
<hr />
<div>Hi, my name is Noor Khan and I am studying Mathematics at Hudson Valley Community College. I am very fond of Mathematics and plan to transfer on to the University at Albany in the fall of 2011, and pursue a bachelors in pure Mathematics. <br />
<br />
My skill level in problem solving is not too high, I'm only able to solve the beginning questions in most of the AMC 12 problems, but I hope that this site will help me achieve higher scores on college Math exams like the NYSMATYC http://www.nysmatyc.org/league/<br />
<br />
I know <math>\LaTeX</math>, so I plan to help out where I see errors on pages (see my contributions). Maybe one day over the summer, I'll add the problems for the NYSMATYC exams I have, with solutions.</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=User:YottaByte&diff=34017User:YottaByte2010-03-26T15:24:46Z<p>YottaByte: </p>
<hr />
<div>Hi, my name is Noor Khan and I am studying Mathematics at Hudson Valley Community College. I am very fond of Mathematics and plan to transfer on to the University at Albany in the fall of 2011, and pursue a bachelors in pure Mathematics. <br />
<br />
My skill level in problem solving is not too high, I'm only able to solve the beginning questions in most of the AMC 12 problems, but I hope that this site will help me achieve higher scores on college Math exams like the NYSMATYC http://www.nysmatyc.org/league/<br />
<br />
I know <math>\LaTeX\</math>, so I plan to help out where I see errors on pages (see my contributions). Maybe one day over the summer, I'll add the problems for the NYSMATYC exams I have, with solutions.</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=User:YottaByte&diff=34016User:YottaByte2010-03-26T15:24:28Z<p>YottaByte: </p>
<hr />
<div>Hi, my name is Noor Khan and I am studying Mathematics at Hudson Valley Community College. I am very fond of Mathematics and plan to transfer on to the University at Albany in the fall of 2011, and pursue a bachelors in pure Mathematics. <br />
<br />
My skill level in problem solving is not too high, I'm only able to solve the beginning questions in most of the AMC 12 problems, but I hope that this site will help me achieve higher scores on college Math exams like the NYSMATYC http://www.nysmatyc.org/league/<br />
<br />
I know $\LaTeX\, so I plan to help out where I see errors on pages (see my contributions). Maybe one day over the summer, I'll add the problems for the NYSMATYC exams I have, with solutions.</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=User:YottaByte&diff=34015User:YottaByte2010-03-26T15:24:13Z<p>YottaByte: </p>
<hr />
<div>Hi, my name is Noor Khan and I am studying Mathematics at Hudson Valley Community College. I am very fond of Mathematics and plan to transfer on to the University at Albany in the fall of 2011, and pursue a bachelors in pure Mathematics. <br />
<br />
My skill level in problem solving is not too high, I'm only able to solve the beginning questions in most of the AMC 12 problems, but I hope that this site will help me achieve higher scores on college Math exams like the NYSMATYC http://www.nysmatyc.org/league/<br />
<br />
I know \LaTeX\, so I plan to help out where I see errors on pages (see my contributions). Maybe one day over the summer, I'll add the problems for the NYSMATYC exams I have, with solutions.</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=User:YottaByte&diff=34014User:YottaByte2010-03-26T15:20:23Z<p>YottaByte: </p>
<hr />
<div>Hi, my name is Noor Khan and I am studying Mathematics at Hudson Valley Community College. I am very fond of Mathematics and plan to transfer on to the University at Albany in the fall of 2011, and pursue a bachelors in pure Mathematics. <br />
<br />
My skill level in problem solving is not too high, I'm only able to solve the beginning questions in most of the AMC 12 problems, but I hope that this site will help me achieve higher scores on college Math exams like the [NYSMATYC http://www.nysmatyc.org/league/].</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=User:YottaByte&diff=33998User:YottaByte2010-03-23T19:25:35Z<p>YottaByte: Created page with 'Hi.'</p>
<hr />
<div>Hi.</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12A_Problems/Problem_13&diff=339972003 AMC 12A Problems/Problem 132010-03-23T19:25:01Z<p>YottaByte: /* Solution */</p>
<hr />
<div>== Problem ==<br />
The [[polygon]] enclosed by the solid lines in the figure consists of 4 [[congruent]] [[square (geometry) | squares]] joined [[edge]]-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a [[cube (geometry) | cube]] with one face missing? <br />
<br />
[[Image:2003amc10a10.gif]]<br />
<br />
<math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 </math><br />
<br />
== Solution ==<br />
[[Image:2003amc10a10.gif]]<br />
<br />
Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>.<br />
<br />
When the polygon is folded, the "right" edge of square <math>A</math> becomes adjacent to the "bottom edge" of square <math>C</math>, and the "bottom" edge of square <math>A</math> becomes adjacent to the "bottom" edge of square <math>D</math>. <br />
<br />
So, any "new" square that is attatched to those edges will prevent the polygon from becoming a cube with one face missing. <br />
<br />
Therefore, squares <math>1</math>, <math>2</math>, and <math>3</math> will prevent the polygon from becoming a cube with one face missing.<br />
<br />
Squares <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, and <math>9</math> will allow the polygon to become a cube with one face missing when folded. <br />
<br />
Thus the answer is <math>6 \Rightarrow E</math>.<br />
<br />
== See Also ==<br />
*[[2003 AMC 12A Problems]]<br />
*[[2003 AMC 12A/Problem 12|Previous Problem]]<br />
*[[2003 AMC 12A/Problem 14|Next Problem]]<br />
<br />
[[Category:Introductory Geometry Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12A_Problems/Problem_13&diff=339962003 AMC 12A Problems/Problem 132010-03-23T19:24:41Z<p>YottaByte: /* Solution */</p>
<hr />
<div>== Problem ==<br />
The [[polygon]] enclosed by the solid lines in the figure consists of 4 [[congruent]] [[square (geometry) | squares]] joined [[edge]]-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a [[cube (geometry) | cube]] with one face missing? <br />
<br />
[[Image:2003amc10a10.gif]]<br />
<br />
<math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 </math><br />
<br />
== Solution ==<br />
{{image}}<br />
[[Image:2003amc10a10.gif]]<br />
<br />
Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>.<br />
<br />
When the polygon is folded, the "right" edge of square <math>A</math> becomes adjacent to the "bottom edge" of square <math>C</math>, and the "bottom" edge of square <math>A</math> becomes adjacent to the "bottom" edge of square <math>D</math>. <br />
<br />
So, any "new" square that is attatched to those edges will prevent the polygon from becoming a cube with one face missing. <br />
<br />
Therefore, squares <math>1</math>, <math>2</math>, and <math>3</math> will prevent the polygon from becoming a cube with one face missing.<br />
<br />
Squares <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, and <math>9</math> will allow the polygon to become a cube with one face missing when folded. <br />
<br />
Thus the answer is <math>6 \Rightarrow E</math>.<br />
<br />
== See Also ==<br />
*[[2003 AMC 12A Problems]]<br />
*[[2003 AMC 12A/Problem 12|Previous Problem]]<br />
*[[2003 AMC 12A/Problem 14|Next Problem]]<br />
<br />
[[Category:Introductory Geometry Problems]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=1951_AHSME_Problems&diff=339891951 AHSME Problems2010-03-22T15:00:53Z<p>YottaByte: /* Problem 2 */</p>
<hr />
<div>== Problem 1 ==<br />
The percent that <math>M</math> is greater than <math>N</math>, is:<br />
<br />
<math> \mathrm{(A) \ } \frac {100(M - N)}{M} \qquad \mathrm{(B) \ } \frac {100(M - N)}{N} \qquad \mathrm{(C) \ } \frac {M - N}{N} \qquad \mathrm{(D) \ } \frac {M - N}{M} \qquad \mathrm{(E) \ } \frac {100(M + N)}{N} </math><br />
<br />
[[1951 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
A rectangular field is half as wide as it is long and is completely enclosed by <math>x</math> yards of fencing. The area in terms of <math>x</math> is:<br />
<br />
<math>(\mathrm{A})\ \frac{x^2}2 \qquad (\mathrm{B})\ 2x^2 \qquad (\mathrm{C})\ \frac{2x^2}9 \qquad (\mathrm{D})\ \frac{x^2}{18} \qquad (\mathrm{E})\ \frac{x^2}{72}</math><br />
<br />
[[1951 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
If the length of a diagonal of a square is <math>a + b</math>, then the area of the square is:<br />
<br />
<math> \mathrm{(A) \ (a+b)^2 } \qquad \mathrm{(B) \ \frac{1}{2}(a+b)^2 } \qquad \mathrm{(C) \ a^2+b^2 } \qquad \mathrm{(D) \ \frac {1}{2}(a^2+b^2) } \qquad \mathrm{(E) \ \text{none of these} } </math><br />
<br />
[[1951 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
A barn with a roof is rectangular in shape, <math>10</math> yd. wide, <math>13</math> yd. long and <math>5</math> yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:<br />
<br />
<math> \mathrm{(A) \ } 360 \qquad \mathrm{(B) \ } 460 \qquad \mathrm{(C) \ } 490 \qquad \mathrm{(D) \ } 590 \qquad \mathrm{(E) \ } 720 </math><br />
<br />
[[1951 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
Mr. <math>A</math> owns a home worth <math>\</math>10,000. He sells it to Mr. <math>B</math> at a 10% profit based on the worth of the house. Mr. <math>B</math> sells the house back to Mr. <math>A</math> at a 10% loss. Then:<br />
<br />
<math> \mathrm{(A) \ A\ comes\ out\ even } \qquad</math> <math>\mathrm{(B) \ A\ makes\ 1100\ on\ the\ deal}</math> <math> \qquad \mathrm{(C) \ A\ makes\ 1000\ on\ the\ deal } \qquad</math> <math>\mathrm{(D) \ A\ loses\ 900\ on\ the\ deal }</math> <math>\qquad \mathrm{(E) \ A\ loses\ 1000\ on\ the\ deal } </math><br />
<br />
[[1951 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
[[195 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
[[1951 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
<br />
[[1951 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
<br />
[[1951 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
[[1951 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[1951 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
[[1951 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[1951 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[1951 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
[[1951 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
If in applying the quadratic formula to a quadratic equation<br />
<br />
<cmath>f(x) \equiv ax^2 + bx + c = 0,</cmath><br />
<br />
it happens that <math>c = \frac{b^2}{4a}</math>, then the graph of <math>y = f(x)</math> will certainly:<br />
<br />
<math>\mathrm{(A) \ have\ a\ maximum } \qquad \mathrm{(B) \ have\ a\ minimum} \qquad</math> <math>\mathrm{(C) \ be\ tangent\ to\ the\ xaxis} \qquad</math> <math>\mathrm{(D) \ be\ tangent\ to\ the\ yaxis} \qquad</math> <math>\mathrm{(E) \ lie\ in\ one\ quadrant\ only}</math><br />
<br />
[[1951 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
[[1951 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
[[1951 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
[[1951 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
[[1951 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[1951 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
[[1951 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[1951 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[1951 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[1951 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
<br />
[[1951 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
<br />
[[1951 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
<br />
[[1951 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
<br />
[[1951 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
<br />
[[1951 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[1995 AHSME]]<br />
* [[Mathematics competition resources]]</div>YottaBytehttps://artofproblemsolving.com/wiki/index.php?title=1951_AHSME_Problems&diff=339881951 AHSME Problems2010-03-22T15:00:14Z<p>YottaByte: /* Problem 1 */</p>
<hr />
<div>== Problem 1 ==<br />
The percent that <math>M</math> is greater than <math>N</math>, is:<br />
<br />
<math> \mathrm{(A) \ } \frac {100(M - N)}{M} \qquad \mathrm{(B) \ } \frac {100(M - N)}{N} \qquad \mathrm{(C) \ } \frac {M - N}{N} \qquad \mathrm{(D) \ } \frac {M - N}{M} \qquad \mathrm{(E) \ } \frac {100(M + N)}{N} </math><br />
<br />
[[1951 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
The percent that <math>M</math> is greater than <math>N</math>, is:<br />
<br />
<math> \mathrm{(A) \ } \frac {100(M - N)}{M} \qquad \mathrm{(B) \ } \frac {100(M - N)}{N} \qquad \mathrm{(C) \ } \frac {M - N}{N} \qquad \mathrm{(D) \ } \frac {M - N}{M} \qquad \mathrm{(E) \ } \frac {100(M + N)}{N} </math><br />
<br />
[[1951 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
If the length of a diagonal of a square is <math>a + b</math>, then the area of the square is:<br />
<br />
<math> \mathrm{(A) \ (a+b)^2 } \qquad \mathrm{(B) \ \frac{1}{2}(a+b)^2 } \qquad \mathrm{(C) \ a^2+b^2 } \qquad \mathrm{(D) \ \frac {1}{2}(a^2+b^2) } \qquad \mathrm{(E) \ \text{none of these} } </math><br />
<br />
[[1951 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
A barn with a roof is rectangular in shape, <math>10</math> yd. wide, <math>13</math> yd. long and <math>5</math> yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:<br />
<br />
<math> \mathrm{(A) \ } 360 \qquad \mathrm{(B) \ } 460 \qquad \mathrm{(C) \ } 490 \qquad \mathrm{(D) \ } 590 \qquad \mathrm{(E) \ } 720 </math><br />
<br />
[[1951 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
Mr. <math>A</math> owns a home worth <math>\</math>10,000. He sells it to Mr. <math>B</math> at a 10% profit based on the worth of the house. Mr. <math>B</math> sells the house back to Mr. <math>A</math> at a 10% loss. Then:<br />
<br />
<math> \mathrm{(A) \ A\ comes\ out\ even } \qquad</math> <math>\mathrm{(B) \ A\ makes\ 1100\ on\ the\ deal}</math> <math> \qquad \mathrm{(C) \ A\ makes\ 1000\ on\ the\ deal } \qquad</math> <math>\mathrm{(D) \ A\ loses\ 900\ on\ the\ deal }</math> <math>\qquad \mathrm{(E) \ A\ loses\ 1000\ on\ the\ deal } </math><br />
<br />
[[1951 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
[[195 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
[[1951 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
<br />
[[1951 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
<br />
[[1951 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
[[1951 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
[[1951 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
[[1951 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
[[1951 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
[[1951 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
[[1951 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
If in applying the quadratic formula to a quadratic equation<br />
<br />
<cmath>f(x) \equiv ax^2 + bx + c = 0,</cmath><br />
<br />
it happens that <math>c = \frac{b^2}{4a}</math>, then the graph of <math>y = f(x)</math> will certainly:<br />
<br />
<math>\mathrm{(A) \ have\ a\ maximum } \qquad \mathrm{(B) \ have\ a\ minimum} \qquad</math> <math>\mathrm{(C) \ be\ tangent\ to\ the\ xaxis} \qquad</math> <math>\mathrm{(D) \ be\ tangent\ to\ the\ yaxis} \qquad</math> <math>\mathrm{(E) \ lie\ in\ one\ quadrant\ only}</math><br />
<br />
[[1951 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
[[1951 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
[[1951 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
[[1951 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
[[1951 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[1951 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
[[1951 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[1951 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[1951 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[1951 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
<br />
[[1951 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
<br />
[[1951 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
<br />
[[1951 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
<br />
[[1951 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
<br />
[[1951 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[1995 AHSME]]<br />
* [[Mathematics competition resources]]</div>YottaByte