https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Freddylukai&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T12:04:24ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=User:Freddylukai&diff=45334User:Freddylukai2012-03-09T01:36:44Z<p>Freddylukai: Created page with "Hey! I'm Kai Lu, but I prefer to be called Freddy. Math is fun! http://www.ticalc.org/archives/files/fileinfo/446/44602.html (Everyone should totally download this program :)..."</p>
<hr />
<div>Hey! I'm Kai Lu, but I prefer to be called Freddy. <br />
<br />
Math is fun!<br />
<br />
http://www.ticalc.org/archives/files/fileinfo/446/44602.html<br />
<br />
(Everyone should totally download this program :) )<br />
<br />
<br />
-thisUnkn0wn</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems/Problem_16&diff=453081985 AHSME Problems/Problem 162012-03-07T02:36:23Z<p>Freddylukai: /* Solution */</p>
<hr />
<div>==Problem==<br />
If <math> A=20^\circ </math> and <math> B=25^\circ </math>, then the value of <math> (1+\tan A)(1+\tan B) </math> is<br />
<br />
<math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2(\tan A+\tan B) \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
==Solution==<br />
===Solution 1===<br />
First, let's leave everything in variables and see if we can simplify <math> (1+\tan A)(1+\tan B) </math>.<br />
<br />
<br />
We can write everything in terms of sine and cosine to get <math> \left(\frac{\cos A}{\cos A}+\frac{\sin A}{\cos A}\right)\left(\frac{\cos B}{\cos B}+\frac{\sin B}{\cos B}\right)=\frac{(\sin A+\cos A)(\sin B+\cos B)}{\cos A\cos B} </math>.<br />
<br />
<br />
<br />
We can multiply out the numerator to get <math> \frac{\sin A\sin B+\cos A\cos B+\sin A\cos B+\sin B\cos A}{\cos A\cos B} </math>.<br />
<br />
<br />
It may seem at first that we've made everything more complicated, however, we can recognize the numerator from the angle sum formulas:<br />
<br />
<br />
<math> \cos(A-B)=\sin A\sin B+\cos A\cos B </math><br />
<br />
<math> \sin(A+B)=\sin A\cos B+\sin B\cos A </math><br />
<br />
<br />
Therefore, our fraction is equal to <math> \frac{\cos(A-B)+\sin(A+B)}{\cos A\cos B} </math>.<br />
<br />
<br />
We can also use the product-to-sum formula<br />
<br />
<math> \cos A\cos B=\frac{1}{2}(\cos(A-B)+\cos(A+B)) </math> to simplify the denominator:<br />
<br />
<br />
<math> \frac{\cos(A-B)+\sin(A+B)}{\frac{1}{2}(\cos(A-B)+\cos(A+B))} </math>.<br />
<br />
<br />
But now we seem stuck. However, we can note that since <math> A+B=45^\circ </math>, we have <math> \sin(A+B)=\cos(A+B) </math>, so we get<br />
<br />
<br />
<math> \frac{\cos(A-B)+\sin(A+B)}{\frac{1}{2}(\cos(A-B)+\sin(A+B))} </math><br />
<br />
<br />
<math> \frac{1}{\frac{1}{2}} </math><br />
<br />
<math> 2, \boxed{\text{B}} </math><br />
<br />
Note that we only used the fact that <math> \sin(A+B)=\cos(A+B) </math>, so we have in fact not just shown that <math> (1+\tan A)(1+\tan B)=2 </math> for <math> A=20^\circ </math> and <math> B=25^\circ </math>, but for all <math> A, B </math> such that <math> A+B=45^\circ+n180^\circ </math>, for integer <math> n </math>.<br />
<br />
<br />
===Solution 2===<br />
<br />
We can see that <math>25^o+20^o=45^o</math>. We also know that <math>\tan 45=1</math>. First, let us expand <math>(1+\tan A)(1+\tan B)</math>.<br />
<br />
We get <math>1+\tan A+\tan B+\tan A\tan B</math>. <br />
<br />
Now, let us look at <math>\tan45=\tan(20+25)</math>.<br />
<br />
By the <math>\tan</math> sum formula, we know that <math>\tan45=\dfrac{\text{tan A}+\text{tan B}}{1- \text{tan A} \text{tan B}}</math><br />
<br />
Then, since <math>\tan 45=1</math>, we can see that <math>\tan A+\tan B=1-\tan A\tan B</math><br />
<br />
Then <math>1=\tan A+\tan B+\tan A\tan B</math><br />
<br />
Thus, the sum become <math>1+1=2</math> and the answer is <math>\fbox{\text{(B)}}</math><br />
<br />
==See Also==<br />
{{AHSME box|year=1985|num-b=15|num-a=17}}</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems/Problem_16&diff=453071985 AHSME Problems/Problem 162012-03-07T02:35:51Z<p>Freddylukai: /* Alternate Solution */</p>
<hr />
<div>==Problem==<br />
If <math> A=20^\circ </math> and <math> B=25^\circ </math>, then the value of <math> (1+\tan A)(1+\tan B) </math> is<br />
<br />
<math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2(\tan A+\tan B) \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
==Solution==<br />
First, let's leave everything in variables and see if we can simplify <math> (1+\tan A)(1+\tan B) </math>.<br />
<br />
<br />
We can write everything in terms of sine and cosine to get <math> \left(\frac{\cos A}{\cos A}+\frac{\sin A}{\cos A}\right)\left(\frac{\cos B}{\cos B}+\frac{\sin B}{\cos B}\right)=\frac{(\sin A+\cos A)(\sin B+\cos B)}{\cos A\cos B} </math>.<br />
<br />
<br />
<br />
We can multiply out the numerator to get <math> \frac{\sin A\sin B+\cos A\cos B+\sin A\cos B+\sin B\cos A}{\cos A\cos B} </math>.<br />
<br />
<br />
It may seem at first that we've made everything more complicated, however, we can recognize the numerator from the angle sum formulas:<br />
<br />
<br />
<math> \cos(A-B)=\sin A\sin B+\cos A\cos B </math><br />
<br />
<math> \sin(A+B)=\sin A\cos B+\sin B\cos A </math><br />
<br />
<br />
Therefore, our fraction is equal to <math> \frac{\cos(A-B)+\sin(A+B)}{\cos A\cos B} </math>.<br />
<br />
<br />
We can also use the product-to-sum formula<br />
<br />
<math> \cos A\cos B=\frac{1}{2}(\cos(A-B)+\cos(A+B)) </math> to simplify the denominator:<br />
<br />
<br />
<math> \frac{\cos(A-B)+\sin(A+B)}{\frac{1}{2}(\cos(A-B)+\cos(A+B))} </math>.<br />
<br />
<br />
But now we seem stuck. However, we can note that since <math> A+B=45^\circ </math>, we have <math> \sin(A+B)=\cos(A+B) </math>, so we get<br />
<br />
<br />
<math> \frac{\cos(A-B)+\sin(A+B)}{\frac{1}{2}(\cos(A-B)+\sin(A+B))} </math><br />
<br />
<br />
<math> \frac{1}{\frac{1}{2}} </math><br />
<br />
<math> 2, \boxed{\text{B}} </math><br />
<br />
Note that we only used the fact that <math> \sin(A+B)=\cos(A+B) </math>, so we have in fact not just shown that <math> (1+\tan A)(1+\tan B)=2 </math> for <math> A=20^\circ </math> and <math> B=25^\circ </math>, but for all <math> A, B </math> such that <math> A+B=45^\circ+n180^\circ </math>, for integer <math> n </math>.<br />
<br />
<br />
===Alternate Solution ===<br />
<br />
We can see that <math>25^o+20^o=45^o</math>. We also know that <math>\tan 45=1</math>. First, let us expand <math>(1+\tan A)(1+\tan B)</math>.<br />
<br />
We get <math>1+\tan A+\tan B+\tan A\tan B</math>. <br />
<br />
Now, let us look at <math>\tan45=\tan(20+25)</math>.<br />
<br />
By the <math>\tan</math> sum formula, we know that <math>\tan45=\dfrac{\text{tan A}+\text{tan B}}{1- \text{tan A} \text{tan B}}</math><br />
<br />
Then, since <math>\tan 45=1</math>, we can see that <math>\tan A+\tan B=1-\tan A\tan B</math><br />
<br />
Then <math>1=\tan A+\tan B+\tan A\tan B</math><br />
<br />
Thus, the sum become <math>1+1=2</math> and the answer is <math>\fbox{\text{(B)}}</math><br />
<br />
==See Also==<br />
{{AHSME box|year=1985|num-b=15|num-a=17}}</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1999_AHSME_Problems&diff=447891999 AHSME Problems2012-02-17T01:58:50Z<p>Freddylukai: /* Problem 28 */</p>
<hr />
<div>== Problem 1 ==<br />
<math>1 - 2 + 3 -4 + \cdots - 98 + 99 = </math><br />
<br />
<math> \mathrm{(A) \ -50 } \qquad \mathrm{(B) \ -49 } \qquad \mathrm{(C) \ 0 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 50 } </math><br />
<br />
[[1999 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
Which of the following statements is false?<br />
<br />
<math> \mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}</math><br />
<math>\mathrm{(B) \ All\ equilateral\ triangles\ are\ convex.}</math><br />
<math>\mathrm{(C) \ All\ equilateral\ triangles\ are\ equiangular.}</math><br />
<math>\mathrm{(D) \ All\ equilateral\ triangles\ are\ regular\ polygons.}</math><br />
<math>\mathrm{(E) \ All\ equilateral\ triangles\ are\ similar\ to\ each\ other.} </math><br />
<br />
[[1999 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The number halfway between <math>1/8</math> and <math>1/10</math> is <br />
<br />
<math> \mathrm{(A) \ } \frac 1{80} \qquad \mathrm{(B) \ } \frac 1{40} \qquad \mathrm{(C) \ } \frac 1{18} \qquad \mathrm{(D) \ } \frac 1{9} \qquad \mathrm{(E) \ } \frac 9{80} </math><br />
<br />
[[1999 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
Find the sum of all prime numbers between <math>1</math> and <math>100</math> that are simultaneously <math>1</math> greater than a multiple of <math>4</math> and <math>1</math> less than a multiple of <math>5</math>. <br />
<br />
<math> \mathrm{(A) \ } 118 \qquad \mathrm{(B) \ }137 \qquad \mathrm{(C) \ } 158 \qquad \mathrm{(D) \ } 187 \qquad \mathrm{(E) \ } 245</math><br />
<br />
[[1999 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
The marked price of a book was <math>30 \%</math> less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?<br />
<br />
<math> \mathrm{(A) \ }25 \% \qquad \mathrm{(B) \ }30 \% \qquad \mathrm{(C) \ }35 \% \qquad \mathrm{(D) \ }60 \% \qquad \mathrm{(E) \ }65 \% </math><br />
<br />
[[1999 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
What is the sum of the digits of the decimal form of the product <math>2^{1999} \cdot 5^{2001}</math>?<br />
<br />
<math> \mathrm{(A) \ }2 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }10 </math><br />
<br />
[[1999 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the largest number of acute angles that a convex hexagon can have? <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 />
[[1999 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
At the end of 1994 Walter was half as old as his grandmother. The sum of the years in which they were born is 3838. How old will Walter be at the end of 1999?<br />
<br />
<math> \mathrm{(A) \ } 48 \qquad \mathrm{(B) \ }49 \qquad \mathrm{(C) \ }53 \qquad \mathrm{(D) \ }55 \qquad \mathrm{(E) \ } 101</math><br />
<br />
[[1999 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of 75 miles per hour, which of the following was her greatest possible average speed?<br />
<br />
<math> \mathrm{(A) \ } 33\frac 13 \qquad \mathrm{(B) \ }53\frac 13 \qquad \mathrm{(C) \ }66\frac 23 \qquad \mathrm{(D) \ }70\frac 13 \qquad \mathrm{(E) \ } 74\frac 13</math><br />
<br />
[[1999 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.<br />
<br />
I. The digit is 1.<br />
<br />
II. the digit is not 2. <br />
<br />
III. The digit is 3.<br />
<br />
IV. The digit is not 4.<br />
<br />
Which one of the following must necessarily be correct?<br />
<br />
<math> \mathrm{(A) \ I\ is\ true} \qquad \mathrm{(B) \ I\ is\ false} \qquad \mathrm{(C) \ II\ is\ true} \qquad \mathrm{(D) \ III\ is\ true} \qquad \mathrm{(E) \ IV\ is\ false} </math><br />
<br />
[[1999 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
The student lockers at Olymmpic High are numbered consecutively beginning with locker number <math>1</math>. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number <math>9</math> and four cents to label locker number <math>10</math>. If it costs <math>\</math> <math> 137.94</math> to label all the lockers, how many lockers are there at the school?<br />
<br />
<math> \mathrm{(A) \ }2001 \qquad \mathrm{(B) \ }2010 \qquad \mathrm{(C) \ }2100 \qquad \mathrm{(D) \ }2726 \qquad \mathrm{(E) \ }6897 </math><br />
<br />
[[1999 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions <math>y = p(x)</math> and <math>y = q(x)</math>, each with leading coefficient <math>1</math>?<br />
<br />
<math>\textrm{(A)} \ 1 \qquad \textrm{(B)} \ 2 \qquad \textrm{(C)} \ 3 \qquad \textrm{(D)} \ 4 \qquad \textrm{(E)} \ 8</math><br />
<br />
[[1999 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
Define a sequence of real numbers <math>a_1, a_2, a_3, \ldots</math> by <math>a_1 = 1</math> and <math>a_{n+1}^3 = 99a_n^3</math> for all <math>n \ge 1</math>. Then <math>a_{100}</math> equals<br />
<br />
<math> \mathrm{(A) \ } 33^{33} \qquad \mathrm{(B) \ } 33^{99} \qquad \mathrm{(C) \ } 99^{33} \qquad \mathrm{(D) \ }99^{99} \qquad \mathrm{(E) \ none\ of\ the\ above} </math><br />
<br />
[[1999 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
Four girls - Mary, Aline, Tina, and Hana - sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 7 songs, which was more than any other girl, and Mary sang 4 songs, which was fewer than any other girl. How many songs did these trios sing? <br />
<br />
<math> \mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 11 } </math><br />
<br />
[[1999 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Let <math>x</math> be a real number such that <math>\sec x - \tan x = 2</math>. Then <math>\sec x + \tan x =</math><br />
<br />
<math> \mathrm{(A) \ } 0.1 \qquad \mathrm{(B) \ } 0.2 \qquad \mathrm{(C) \ } 0.3 \qquad \mathrm{(D) \ } 0.4 \qquad \mathrm{(E) \ } 0.5</math><br />
<br />
[[1999 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
What is the radius of a circle inscribed in a rhombus with diagonals of length <math>10</math> and <math>24</math>?<br />
<br />
<math> \mathrm{(A) \ }4 \qquad \mathrm{(B) \ }\frac {58}{13} \qquad \mathrm{(C) \ }\frac{60}{13} \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6 </math><br />
<br />
[[1999 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Let <math>P(x)</math> be a polynomial such that when <math>P(x)</math> is divided by <math>x-19</math>, the remainder is <math>99</math>, and when <math>P(x)</math> is divided by <math>x - 99</math>, the remainder is <math>19</math>. What is the remainder when <math>P(x)</math> is divided by <math>(x-19)(x-99)</math>?<br />
<br />
<math> \mathrm{(A) \ } -x + 80 \qquad \mathrm{(B) \ } x + 80 \qquad \mathrm{(C) \ } -x + 118 \qquad \mathrm{(D) \ } x + 118 \qquad \mathrm{(E) \ } 0</math><br />
<br />
[[1999 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
How many zeros does <math>f(x) = \cos(\log x)</math> have on the interval <math>0 < x < 1</math>?<br />
<br />
<math> \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } \text{infinitely\ many}</math><br />
<br />
[[1999 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Consider all triangles <math>ABC</math> satisfying in the following conditions: <math>AB = AC</math>, <math>D</math> is a point on <math>\overline{AC}</math> for which <math>\overline{BD} \perp \overline{AC}</math>, <math>AC</math> and <math>CD</math> are integers, and <math>BD^{2} = 57</math>. Among all such triangles, the smallest possible value of <math>AC</math> is<br />
<br />
<asy><br />
pair A,B,C,D; <br />
A=(5,12); B=origin; C=(10,0); D=(8.52071005917,3.55029585799);<br />
draw(A--B--C--cycle); draw(B--D);<br />
label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE);<br />
</asy><br />
<br />
<math>\textrm{(A)} \ 9 \qquad \textrm{(B)} \ 10 \qquad \textrm{(C)} \ 11 \qquad \textrm{(D)} \ 12 \qquad \textrm{(E)} \ 13</math> <br />
<br />
[[1999 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
The sequence <math>a_{1},a_{2},a_{3},\ldots</math> satisfies <math>a_{1} = 19,a_{9} = 99</math>, and, for all <math>n\geq 3</math>, <math>a_{n}</math> is the arithmetic mean of the first <math>n - 1</math> terms. Find <math>a_2</math>.<br />
<br />
<math>\textrm{(A)} \ 29 \qquad \textrm{(B)} \ 59 \qquad \textrm{(C)} \ 79 \qquad \textrm{(D)} \ 99 \qquad \textrm{(E)} \ 179</math><br />
<br />
[[1999 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
A circle is circumscribed about a triangle with sides <math>20,21,</math> and <math>29,</math> thus dividing the interior of the circle into four regions. Let <math>A,B,</math> and <math>C</math> be the areas of the non-triangular regions, with <math>C</math> be the largest. Then<br />
<br />
<math> \mathrm{(A) \ }A+B=C \qquad \mathrm{(B) \ }A+B+210=C \qquad \mathrm{(C) \ }A^2+B^2=C^2 \qquad \mathrm{(D) \ }20A+21B=29C \qquad \mathrm{(E) \ } \frac 1{A^2}+\frac 1{B^2}= \frac 1{C^2}</math><br />
<br />
[[1999 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
The graphs of <math>y = -|x-a| + b</math> and <math>y = |x-c| + d</math> intersect at points <math>(2,5)</math> and <math>(8,3)</math>. Find <math>a+c</math>.<br />
<br />
<math> \mathrm{(A) \ } 7 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ } 13\qquad \mathrm{(E) \ } 18</math><br />
<br />
[[1999 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The equiangular convex hexagon <math>ABCDEF</math> has <math>AB = 1, BC = 4, CD = 2,</math> and <math>DE = 4.</math> The area of the hexagon is <br />
<br />
<math> \mathrm{(A) \ } \frac {15}2\sqrt{3} \qquad \mathrm{(B) \ }9\sqrt{3} \qquad \mathrm{(C) \ }16 \qquad \mathrm{(D) \ }\frac{39}4\sqrt{3} \qquad \mathrm{(E) \ } \frac{43}4\sqrt{3}</math><br />
<br />
[[1999 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral?<br />
<br />
<math> \mathrm{(A) \ } \frac 1{15} \qquad \mathrm{(B) \ } \frac 1{91} \qquad \mathrm{(C) \ } \frac 1{273} \qquad \mathrm{(D) \ } \frac 1{455} \qquad \mathrm{(E) \ } \frac 1{1365}</math><br />
<br />
[[1999 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
There are unique integers <math>a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}</math> such that<br />
<br />
<cmath>\frac {5}{7} = \frac {a_{2}}{2!} + \frac {a_{3}}{3!} + \frac {a_{4}}{4!} + \frac {a_{5}}{5!} + \frac {a_{6}}{6!} + \frac {a_{7}}{7!}</cmath><br />
<br />
where <math>0\leq a_{i} < i</math> for <math>i = 2,3,\ldots,7</math>. Find <math>a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}</math>.<br />
<br />
<math>\textrm{(A)} \ 8 \qquad \textrm{(B)} \ 9 \qquad \textrm{(C)} \ 10 \qquad \textrm{(D)} \ 11 \qquad \textrm{(E)} \ 12</math><br />
<br />
[[1999 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length <math>1</math>. The polygons meet at a point <math>A</math> in such a way that the sum of the three interior angles at <math>A</math> is <math>360^{\circ}</math>. Thus the three polygons form a new polygon with <math>A</math> as an interior point. What is the largest possible perimeter that this polygon can have? <br />
<br />
<math> \mathrm{(A) \ }12 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ }18 \qquad \mathrm{(D) \ }21 \qquad \mathrm{(E) \ } 24</math><br />
<br />
[[1999 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
In triangle <math>ABC</math>, <math>3 \sin A + 4 \cos B = 6</math> and <math>4 \sin B + 3 \cos A = 1</math>. Then <math>\angle C</math> in degrees is <br />
<br />
<math> \mathrm{(A) \ }30 \qquad \mathrm{(B) \ }60 \qquad \mathrm{(C) \ }90 \qquad \mathrm{(D) \ }120 \qquad \mathrm{(E) \ }150 </math><br />
<br />
[[1999 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Let <math>x_1, x_2, \ldots , x_n</math> be a sequence of integers such that<br />
<br />
<math>\text{(i)}</math> <math>-1 \le x_i \le 2</math> <math>\text{for}</math> <math>i = 1,2, \ldots n</math><br />
<br />
<math>\text{(ii)}</math> <math>x_1 + \cdots + x_n = 19</math>; <math>\text{and}</math><br />
<br />
<math>\text{(iii)}</math> <math>x_1^2 + x_2^2 + \cdots + x_n^2 = 99</math>.<br />
<br />
Let <math>m</math> and <math>M</math> be the minimal and maximal possible values of <math>x_1^3 + \cdots + x_n^3</math>, respectively. Then <math>\frac Mm =</math><br />
<br />
<math> \mathrm{(A) \ }3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ }7 </math><br />
<br />
[[1999 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. a point <math>P</math> is selected at random inside the circumscribed sphere. The probability that <math>P</math> lies inside one of the five small spheres is closest to <br />
<br />
<math> \mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4 </math><br />
<br />
[[1999 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
The number of ordered pairs of integers <math>(m,n)</math> for which <math>mn \ge 0</math> and<br />
<br />
<cmath>m^3 + n^3 + 99mn = 33^3</cmath><br />
<br />
is equal to<br />
<br />
<math> \mathrm{(A) \ }2 \qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ }35 \qquad \mathrm{(E) \ } 99</math><br />
<br />
[[1999 AHSME Problems/Problem 30|Solution]]<br />
<br />
<br />
== See also ==<br />
*[[AHSME]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1999_AHSME_Problems&diff=447881999 AHSME Problems2012-02-17T01:52:08Z<p>Freddylukai: /* Problem 28 */</p>
<hr />
<div>== Problem 1 ==<br />
<math>1 - 2 + 3 -4 + \cdots - 98 + 99 = </math><br />
<br />
<math> \mathrm{(A) \ -50 } \qquad \mathrm{(B) \ -49 } \qquad \mathrm{(C) \ 0 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 50 } </math><br />
<br />
[[1999 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
Which of the following statements is false?<br />
<br />
<math> \mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}</math><br />
<math>\mathrm{(B) \ All\ equilateral\ triangles\ are\ convex.}</math><br />
<math>\mathrm{(C) \ All\ equilateral\ triangles\ are\ equiangular.}</math><br />
<math>\mathrm{(D) \ All\ equilateral\ triangles\ are\ regular\ polygons.}</math><br />
<math>\mathrm{(E) \ All\ equilateral\ triangles\ are\ similar\ to\ each\ other.} </math><br />
<br />
[[1999 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The number halfway between <math>1/8</math> and <math>1/10</math> is <br />
<br />
<math> \mathrm{(A) \ } \frac 1{80} \qquad \mathrm{(B) \ } \frac 1{40} \qquad \mathrm{(C) \ } \frac 1{18} \qquad \mathrm{(D) \ } \frac 1{9} \qquad \mathrm{(E) \ } \frac 9{80} </math><br />
<br />
[[1999 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
Find the sum of all prime numbers between <math>1</math> and <math>100</math> that are simultaneously <math>1</math> greater than a multiple of <math>4</math> and <math>1</math> less than a multiple of <math>5</math>. <br />
<br />
<math> \mathrm{(A) \ } 118 \qquad \mathrm{(B) \ }137 \qquad \mathrm{(C) \ } 158 \qquad \mathrm{(D) \ } 187 \qquad \mathrm{(E) \ } 245</math><br />
<br />
[[1999 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
The marked price of a book was <math>30 \%</math> less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?<br />
<br />
<math> \mathrm{(A) \ }25 \% \qquad \mathrm{(B) \ }30 \% \qquad \mathrm{(C) \ }35 \% \qquad \mathrm{(D) \ }60 \% \qquad \mathrm{(E) \ }65 \% </math><br />
<br />
[[1999 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
What is the sum of the digits of the decimal form of the product <math>2^{1999} \cdot 5^{2001}</math>?<br />
<br />
<math> \mathrm{(A) \ }2 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }10 </math><br />
<br />
[[1999 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the largest number of acute angles that a convex hexagon can have? <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 />
[[1999 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
At the end of 1994 Walter was half as old as his grandmother. The sum of the years in which they were born is 3838. How old will Walter be at the end of 1999?<br />
<br />
<math> \mathrm{(A) \ } 48 \qquad \mathrm{(B) \ }49 \qquad \mathrm{(C) \ }53 \qquad \mathrm{(D) \ }55 \qquad \mathrm{(E) \ } 101</math><br />
<br />
[[1999 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of 75 miles per hour, which of the following was her greatest possible average speed?<br />
<br />
<math> \mathrm{(A) \ } 33\frac 13 \qquad \mathrm{(B) \ }53\frac 13 \qquad \mathrm{(C) \ }66\frac 23 \qquad \mathrm{(D) \ }70\frac 13 \qquad \mathrm{(E) \ } 74\frac 13</math><br />
<br />
[[1999 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.<br />
<br />
I. The digit is 1.<br />
<br />
II. the digit is not 2. <br />
<br />
III. The digit is 3.<br />
<br />
IV. The digit is not 4.<br />
<br />
Which one of the following must necessarily be correct?<br />
<br />
<math> \mathrm{(A) \ I\ is\ true} \qquad \mathrm{(B) \ I\ is\ false} \qquad \mathrm{(C) \ II\ is\ true} \qquad \mathrm{(D) \ III\ is\ true} \qquad \mathrm{(E) \ IV\ is\ false} </math><br />
<br />
[[1999 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
The student lockers at Olymmpic High are numbered consecutively beginning with locker number <math>1</math>. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number <math>9</math> and four cents to label locker number <math>10</math>. If it costs <math>\</math> <math> 137.94</math> to label all the lockers, how many lockers are there at the school?<br />
<br />
<math> \mathrm{(A) \ }2001 \qquad \mathrm{(B) \ }2010 \qquad \mathrm{(C) \ }2100 \qquad \mathrm{(D) \ }2726 \qquad \mathrm{(E) \ }6897 </math><br />
<br />
[[1999 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions <math>y = p(x)</math> and <math>y = q(x)</math>, each with leading coefficient <math>1</math>?<br />
<br />
<math>\textrm{(A)} \ 1 \qquad \textrm{(B)} \ 2 \qquad \textrm{(C)} \ 3 \qquad \textrm{(D)} \ 4 \qquad \textrm{(E)} \ 8</math><br />
<br />
[[1999 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
Define a sequence of real numbers <math>a_1, a_2, a_3, \ldots</math> by <math>a_1 = 1</math> and <math>a_{n+1}^3 = 99a_n^3</math> for all <math>n \ge 1</math>. Then <math>a_{100}</math> equals<br />
<br />
<math> \mathrm{(A) \ } 33^{33} \qquad \mathrm{(B) \ } 33^{99} \qquad \mathrm{(C) \ } 99^{33} \qquad \mathrm{(D) \ }99^{99} \qquad \mathrm{(E) \ none\ of\ the\ above} </math><br />
<br />
[[1999 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
Four girls - Mary, Aline, Tina, and Hana - sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 7 songs, which was more than any other girl, and Mary sang 4 songs, which was fewer than any other girl. How many songs did these trios sing? <br />
<br />
<math> \mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 11 } </math><br />
<br />
[[1999 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Let <math>x</math> be a real number such that <math>\sec x - \tan x = 2</math>. Then <math>\sec x + \tan x =</math><br />
<br />
<math> \mathrm{(A) \ } 0.1 \qquad \mathrm{(B) \ } 0.2 \qquad \mathrm{(C) \ } 0.3 \qquad \mathrm{(D) \ } 0.4 \qquad \mathrm{(E) \ } 0.5</math><br />
<br />
[[1999 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
What is the radius of a circle inscribed in a rhombus with diagonals of length <math>10</math> and <math>24</math>?<br />
<br />
<math> \mathrm{(A) \ }4 \qquad \mathrm{(B) \ }\frac {58}{13} \qquad \mathrm{(C) \ }\frac{60}{13} \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6 </math><br />
<br />
[[1999 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Let <math>P(x)</math> be a polynomial such that when <math>P(x)</math> is divided by <math>x-19</math>, the remainder is <math>99</math>, and when <math>P(x)</math> is divided by <math>x - 99</math>, the remainder is <math>19</math>. What is the remainder when <math>P(x)</math> is divided by <math>(x-19)(x-99)</math>?<br />
<br />
<math> \mathrm{(A) \ } -x + 80 \qquad \mathrm{(B) \ } x + 80 \qquad \mathrm{(C) \ } -x + 118 \qquad \mathrm{(D) \ } x + 118 \qquad \mathrm{(E) \ } 0</math><br />
<br />
[[1999 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
How many zeros does <math>f(x) = \cos(\log x)</math> have on the interval <math>0 < x < 1</math>?<br />
<br />
<math> \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } \text{infinitely\ many}</math><br />
<br />
[[1999 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Consider all triangles <math>ABC</math> satisfying in the following conditions: <math>AB = AC</math>, <math>D</math> is a point on <math>\overline{AC}</math> for which <math>\overline{BD} \perp \overline{AC}</math>, <math>AC</math> and <math>CD</math> are integers, and <math>BD^{2} = 57</math>. Among all such triangles, the smallest possible value of <math>AC</math> is<br />
<br />
<asy><br />
pair A,B,C,D; <br />
A=(5,12); B=origin; C=(10,0); D=(8.52071005917,3.55029585799);<br />
draw(A--B--C--cycle); draw(B--D);<br />
label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE);<br />
</asy><br />
<br />
<math>\textrm{(A)} \ 9 \qquad \textrm{(B)} \ 10 \qquad \textrm{(C)} \ 11 \qquad \textrm{(D)} \ 12 \qquad \textrm{(E)} \ 13</math> <br />
<br />
[[1999 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
The sequence <math>a_{1},a_{2},a_{3},\ldots</math> satisfies <math>a_{1} = 19,a_{9} = 99</math>, and, for all <math>n\geq 3</math>, <math>a_{n}</math> is the arithmetic mean of the first <math>n - 1</math> terms. Find <math>a_2</math>.<br />
<br />
<math>\textrm{(A)} \ 29 \qquad \textrm{(B)} \ 59 \qquad \textrm{(C)} \ 79 \qquad \textrm{(D)} \ 99 \qquad \textrm{(E)} \ 179</math><br />
<br />
[[1999 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
A circle is circumscribed about a triangle with sides <math>20,21,</math> and <math>29,</math> thus dividing the interior of the circle into four regions. Let <math>A,B,</math> and <math>C</math> be the areas of the non-triangular regions, with <math>C</math> be the largest. Then<br />
<br />
<math> \mathrm{(A) \ }A+B=C \qquad \mathrm{(B) \ }A+B+210=C \qquad \mathrm{(C) \ }A^2+B^2=C^2 \qquad \mathrm{(D) \ }20A+21B=29C \qquad \mathrm{(E) \ } \frac 1{A^2}+\frac 1{B^2}= \frac 1{C^2}</math><br />
<br />
[[1999 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
The graphs of <math>y = -|x-a| + b</math> and <math>y = |x-c| + d</math> intersect at points <math>(2,5)</math> and <math>(8,3)</math>. Find <math>a+c</math>.<br />
<br />
<math> \mathrm{(A) \ } 7 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ } 13\qquad \mathrm{(E) \ } 18</math><br />
<br />
[[1999 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The equiangular convex hexagon <math>ABCDEF</math> has <math>AB = 1, BC = 4, CD = 2,</math> and <math>DE = 4.</math> The area of the hexagon is <br />
<br />
<math> \mathrm{(A) \ } \frac {15}2\sqrt{3} \qquad \mathrm{(B) \ }9\sqrt{3} \qquad \mathrm{(C) \ }16 \qquad \mathrm{(D) \ }\frac{39}4\sqrt{3} \qquad \mathrm{(E) \ } \frac{43}4\sqrt{3}</math><br />
<br />
[[1999 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral?<br />
<br />
<math> \mathrm{(A) \ } \frac 1{15} \qquad \mathrm{(B) \ } \frac 1{91} \qquad \mathrm{(C) \ } \frac 1{273} \qquad \mathrm{(D) \ } \frac 1{455} \qquad \mathrm{(E) \ } \frac 1{1365}</math><br />
<br />
[[1999 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
There are unique integers <math>a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}</math> such that<br />
<br />
<cmath>\frac {5}{7} = \frac {a_{2}}{2!} + \frac {a_{3}}{3!} + \frac {a_{4}}{4!} + \frac {a_{5}}{5!} + \frac {a_{6}}{6!} + \frac {a_{7}}{7!}</cmath><br />
<br />
where <math>0\leq a_{i} < i</math> for <math>i = 2,3,\ldots,7</math>. Find <math>a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}</math>.<br />
<br />
<math>\textrm{(A)} \ 8 \qquad \textrm{(B)} \ 9 \qquad \textrm{(C)} \ 10 \qquad \textrm{(D)} \ 11 \qquad \textrm{(E)} \ 12</math><br />
<br />
[[1999 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length <math>1</math>. The polygons meet at a point <math>A</math> in such a way that the sum of the three interior angles at <math>A</math> is <math>360^{\circ}</math>. Thus the three polygons form a new polygon with <math>A</math> as an interior point. What is the largest possible perimeter that this polygon can have? <br />
<br />
<math> \mathrm{(A) \ }12 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ }18 \qquad \mathrm{(D) \ }21 \qquad \mathrm{(E) \ } 24</math><br />
<br />
[[1999 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
In triangle <math>ABC</math>, <math>3 \sin A + 4 \cos B = 6</math> and <math>4 \sin B + 3 \cos A = 1</math>. Then <math>\angle C</math> in degrees is <br />
<br />
<math> \mathrm{(A) \ }30 \qquad \mathrm{(B) \ }60 \qquad \mathrm{(C) \ }90 \qquad \mathrm{(D) \ }120 \qquad \mathrm{(E) \ }150 </math><br />
<br />
[[1999 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Let <math>x_1, x_2, \ldots , x_n</math> be a sequence of integers such that<br />
<br />
<math>\text{(i)}</math> <math>-1 \le x_i \le 2</math> \text{for} <math>i = 1,2, \ldots n</math><br />
<br />
<math>\text{(ii)}</math> <math>x_1 + \cdots + x_n = 19</math>; <math>\text{and}</math><br />
<br />
<math>\text{(iii)}</math> <math>x_1^2 + x_2^2 + \cdots + x_n^2 = 99</math>.<br />
<br />
Let <math>m</math> and <math>M</math> be the minimal and maximal possible values of <math>x_1^3 + \cdots + x_n^3</math>, respectively. Then <math>\frac Mm =</math><br />
<br />
<math> \mathrm{(A) \ }3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ }7 </math><br />
<br />
[[1999 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. a point <math>P</math> is selected at random inside the circumscribed sphere. The probability that <math>P</math> lies inside one of the five small spheres is closest to <br />
<br />
<math> \mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4 </math><br />
<br />
[[1999 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
The number of ordered pairs of integers <math>(m,n)</math> for which <math>mn \ge 0</math> and<br />
<br />
<cmath>m^3 + n^3 + 99mn = 33^3</cmath><br />
<br />
is equal to<br />
<br />
<math> \mathrm{(A) \ }2 \qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ }35 \qquad \mathrm{(E) \ } 99</math><br />
<br />
[[1999 AHSME Problems/Problem 30|Solution]]<br />
<br />
<br />
== See also ==<br />
*[[AHSME]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1999_AHSME_Problems&diff=447861999 AHSME Problems2012-02-16T23:03:42Z<p>Freddylukai: /* Problem 23 */</p>
<hr />
<div>== Problem 1 ==<br />
<math>1 - 2 + 3 -4 + \cdots - 98 + 99 = </math><br />
<br />
<math> \mathrm{(A) \ -50 } \qquad \mathrm{(B) \ -49 } \qquad \mathrm{(C) \ 0 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 50 } </math><br />
<br />
[[1999 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
Which of the following statements is false?<br />
<br />
<math> \mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}</math><br />
<math>\mathrm{(B) \ All\ equilateral\ triangles\ are\ convex.}</math><br />
<math>\mathrm{(C) \ All\ equilateral\ triangles\ are\ equiangular.}</math><br />
<math>\mathrm{(D) \ All\ equilateral\ triangles\ are\ regular\ polygons.}</math><br />
<math>\mathrm{(E) \ All\ equilateral\ triangles\ are\ similar\ to\ each\ other.} </math><br />
<br />
[[1999 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The number halfway between <math>1/8</math> and <math>1/10</math> is <br />
<br />
<math> \mathrm{(A) \ } \frac 1{80} \qquad \mathrm{(B) \ } \frac 1{40} \qquad \mathrm{(C) \ } \frac 1{18} \qquad \mathrm{(D) \ } \frac 1{9} \qquad \mathrm{(E) \ } \frac 9{80} </math><br />
<br />
[[1999 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
Find the sum of all prime numbers between <math>1</math> and <math>100</math> that are simultaneously <math>1</math> greater than a multiple of <math>4</math> and <math>1</math> less than a multiple of <math>5</math>. <br />
<br />
<math> \mathrm{(A) \ } 118 \qquad \mathrm{(B) \ }137 \qquad \mathrm{(C) \ } 158 \qquad \mathrm{(D) \ } 187 \qquad \mathrm{(E) \ } 245</math><br />
<br />
[[1999 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
The marked price of a book was <math>30 \%</math> less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?<br />
<br />
<math> \mathrm{(A) \ }25 \% \qquad \mathrm{(B) \ }30 \% \qquad \mathrm{(C) \ }35 \% \qquad \mathrm{(D) \ }60 \% \qquad \mathrm{(E) \ }65 \% </math><br />
<br />
[[1999 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
What is the sum of the digits of the decimal form of the product <math>2^{1999} \cdot 5^{2001}</math>?<br />
<br />
<math> \mathrm{(A) \ }2 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }10 </math><br />
<br />
[[1999 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
What is the largest number of acute angles that a convex hexagon can have? <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 />
[[1999 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
At the end of 1994 Walter was half as old as his grandmother. The sum of the years in which they were born is 3838. How old will Walter be at the end of 1999?<br />
<br />
<math> \mathrm{(A) \ } 48 \qquad \mathrm{(B) \ }49 \qquad \mathrm{(C) \ }53 \qquad \mathrm{(D) \ }55 \qquad \mathrm{(E) \ } 101</math><br />
<br />
[[1999 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of 75 miles per hour, which of the following was her greatest possible average speed?<br />
<br />
<math> \mathrm{(A) \ } 33\frac 13 \qquad \mathrm{(B) \ }53\frac 13 \qquad \mathrm{(C) \ }66\frac 23 \qquad \mathrm{(D) \ }70\frac 13 \qquad \mathrm{(E) \ } 74\frac 13</math><br />
<br />
[[1999 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.<br />
<br />
I. The digit is 1.<br />
<br />
II. the digit is not 2. <br />
<br />
III. The digit is 3.<br />
<br />
IV. The digit is not 4.<br />
<br />
Which one of the following must necessarily be correct?<br />
<br />
<math> \mathrm{(A) \ I\ is\ true} \qquad \mathrm{(B) \ I\ is\ false} \qquad \mathrm{(C) \ II\ is\ true} \qquad \mathrm{(D) \ III\ is\ true} \qquad \mathrm{(E) \ IV\ is\ false} </math><br />
<br />
[[1999 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
The student lockers at Olymmpic High are numbered consecutively beginning with locker number <math>1</math>. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number <math>9</math> and four cents to label locker number <math>10</math>. If it costs <math>\</math> <math> 137.94</math> to label all the lockers, how many lockers are there at the school?<br />
<br />
<math> \mathrm{(A) \ }2001 \qquad \mathrm{(B) \ }2010 \qquad \mathrm{(C) \ }2100 \qquad \mathrm{(D) \ }2726 \qquad \mathrm{(E) \ }6897 </math><br />
<br />
[[1999 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions <math>y = p(x)</math> and <math>y = q(x)</math>, each with leading coefficient <math>1</math>?<br />
<br />
<math>\textrm{(A)} \ 1 \qquad \textrm{(B)} \ 2 \qquad \textrm{(C)} \ 3 \qquad \textrm{(D)} \ 4 \qquad \textrm{(E)} \ 8</math><br />
<br />
[[1999 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
Define a sequence of real numbers <math>a_1, a_2, a_3, \ldots</math> by <math>a_1 = 1</math> and <math>a_{n+1}^3 = 99a_n^3</math> for all <math>n \ge 1</math>. Then <math>a_{100}</math> equals<br />
<br />
<math> \mathrm{(A) \ } 33^{33} \qquad \mathrm{(B) \ } 33^{99} \qquad \mathrm{(C) \ } 99^{33} \qquad \mathrm{(D) \ }99^{99} \qquad \mathrm{(E) \ none\ of\ the\ above} </math><br />
<br />
[[1999 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
Four girls - Mary, Aline, Tina, and Hana - sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 7 songs, which was more than any other girl, and Mary sang 4 songs, which was fewer than any other girl. How many songs did these trios sing? <br />
<br />
<math> \mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 11 } </math><br />
<br />
[[1999 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Let <math>x</math> be a real number such that <math>\sec x - \tan x = 2</math>. Then <math>\sec x + \tan x =</math><br />
<br />
<math> \mathrm{(A) \ } 0.1 \qquad \mathrm{(B) \ } 0.2 \qquad \mathrm{(C) \ } 0.3 \qquad \mathrm{(D) \ } 0.4 \qquad \mathrm{(E) \ } 0.5</math><br />
<br />
[[1999 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
What is the radius of a circle inscribed in a rhombus with diagonals of length <math>10</math> and <math>24</math>?<br />
<br />
<math> \mathrm{(A) \ }4 \qquad \mathrm{(B) \ }\frac {58}{13} \qquad \mathrm{(C) \ }\frac{60}{13} \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6 </math><br />
<br />
[[1999 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Let <math>P(x)</math> be a polynomial such that when <math>P(x)</math> is divided by <math>x-19</math>, the remainder is <math>99</math>, and when <math>P(x)</math> is divided by <math>x - 99</math>, the remainder is <math>19</math>. What is the remainder when <math>P(x)</math> is divided by <math>(x-19)(x-99)</math>?<br />
<br />
<math> \mathrm{(A) \ } -x + 80 \qquad \mathrm{(B) \ } x + 80 \qquad \mathrm{(C) \ } -x + 118 \qquad \mathrm{(D) \ } x + 118 \qquad \mathrm{(E) \ } 0</math><br />
<br />
[[1999 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
How many zeros does <math>f(x) = \cos(\log x)</math> have on the interval <math>0 < x < 1</math>?<br />
<br />
<math> \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } \text{infinitely\ many}</math><br />
<br />
[[1999 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
Consider all triangles <math>ABC</math> satisfying in the following conditions: <math>AB = AC</math>, <math>D</math> is a point on <math>\overline{AC}</math> for which <math>\overline{BD} \perp \overline{AC}</math>, <math>AC</math> and <math>CD</math> are integers, and <math>BD^{2} = 57</math>. Among all such triangles, the smallest possible value of <math>AC</math> is<br />
<br />
<asy><br />
pair A,B,C,D; <br />
A=(5,12); B=origin; C=(10,0); D=(8.52071005917,3.55029585799);<br />
draw(A--B--C--cycle); draw(B--D);<br />
label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE);<br />
</asy><br />
<br />
<math>\textrm{(A)} \ 9 \qquad \textrm{(B)} \ 10 \qquad \textrm{(C)} \ 11 \qquad \textrm{(D)} \ 12 \qquad \textrm{(E)} \ 13</math> <br />
<br />
[[1999 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
The sequence <math>a_{1},a_{2},a_{3},\ldots</math> satisfies <math>a_{1} = 19,a_{9} = 99</math>, and, for all <math>n\geq 3</math>, <math>a_{n}</math> is the arithmetic mean of the first <math>n - 1</math> terms. Find <math>a_2</math>.<br />
<br />
<math>\textrm{(A)} \ 29 \qquad \textrm{(B)} \ 59 \qquad \textrm{(C)} \ 79 \qquad \textrm{(D)} \ 99 \qquad \textrm{(E)} \ 179</math><br />
<br />
[[1999 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
A circle is circumscribed about a triangle with sides <math>20,21,</math> and <math>29,</math> thus dividing the interior of the circle into four regions. Let <math>A,B,</math> and <math>C</math> be the areas of the non-triangular regions, with <math>C</math> be the largest. Then<br />
<br />
<math> \mathrm{(A) \ }A+B=C \qquad \mathrm{(B) \ }A+B+210=C \qquad \mathrm{(C) \ }A^2+B^2=C^2 \qquad \mathrm{(D) \ }20A+21B=29C \qquad \mathrm{(E) \ } \frac 1{A^2}+\frac 1{B^2}= \frac 1{C^2}</math><br />
<br />
[[1999 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
The graphs of <math>y = -|x-a| + b</math> and <math>y = |x-c| + d</math> intersect at points <math>(2,5)</math> and <math>(8,3)</math>. Find <math>a+c</math>.<br />
<br />
<math> \mathrm{(A) \ } 7 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ } 13\qquad \mathrm{(E) \ } 18</math><br />
<br />
[[1999 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The equiangular convex hexagon <math>ABCDEF</math> has <math>AB = 1, BC = 4, CD = 2,</math> and <math>DE = 4.</math> The area of the hexagon is <br />
<br />
<math> \mathrm{(A) \ } \frac {15}2\sqrt{3} \qquad \mathrm{(B) \ }9\sqrt{3} \qquad \mathrm{(C) \ }16 \qquad \mathrm{(D) \ }\frac{39}4\sqrt{3} \qquad \mathrm{(E) \ } \frac{43}4\sqrt{3}</math><br />
<br />
[[1999 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral?<br />
<br />
<math> \mathrm{(A) \ } \frac 1{15} \qquad \mathrm{(B) \ } \frac 1{91} \qquad \mathrm{(C) \ } \frac 1{273} \qquad \mathrm{(D) \ } \frac 1{455} \qquad \mathrm{(E) \ } \frac 1{1365}</math><br />
<br />
[[1999 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
There are unique integers <math>a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}</math> such that<br />
<br />
<cmath>\frac {5}{7} = \frac {a_{2}}{2!} + \frac {a_{3}}{3!} + \frac {a_{4}}{4!} + \frac {a_{5}}{5!} + \frac {a_{6}}{6!} + \frac {a_{7}}{7!}</cmath><br />
<br />
where <math>0\leq a_{i} < i</math> for <math>i = 2,3,\ldots,7</math>. Find <math>a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}</math>.<br />
<br />
<math>\textrm{(A)} \ 8 \qquad \textrm{(B)} \ 9 \qquad \textrm{(C)} \ 10 \qquad \textrm{(D)} \ 11 \qquad \textrm{(E)} \ 12</math><br />
<br />
[[1999 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length <math>1</math>. The polygons meet at a point <math>A</math> in such a way that the sum of the three interior angles at <math>A</math> is <math>360^{\circ}</math>. Thus the three polygons form a new polygon with <math>A</math> as an interior point. What is the largest possible perimeter that this polygon can have? <br />
<br />
<math> \mathrm{(A) \ }12 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ }18 \qquad \mathrm{(D) \ }21 \qquad \mathrm{(E) \ } 24</math><br />
<br />
[[1999 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
In triangle <math>ABC</math>, <math>3 \sin A + 4 \cos B = 6</math> and <math>4 \sin B + 3 \cos A = 1</math>. Then <math>\angle C</math> in degrees is <br />
<br />
<math> \mathrm{(A) \ }30 \qquad \mathrm{(B) \ }60 \qquad \mathrm{(C) \ }90 \qquad \mathrm{(D) \ }120 \qquad \mathrm{(E) \ }150 </math><br />
<br />
[[1999 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Let <math>x_1, x_2, \ldots , x_n</math> be a sequence of integers such that<br />
(i) <math>-1 \le x_i \le 2</math> for <math>i = 1,2, \ldots n</math><br />
(ii) <math>x_1 + \cdots + x_n = 19</math>; and<br />
(iii) <math>x_1^2 + x_2^2 + \cdots + x_n^2 = 99</math>.<br />
Let <math>m</math> and <math>M</math> be the minimal and maximal possible values of <math>x_1^3 + \cdots + x_n^3</math>, respectively. Then <math>\frac Mm =</math><br />
<br />
<math> \mathrm{(A) \ }3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ }7 </math><br />
<br />
[[1999 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. a point <math>P</math> is selected at random inside the circumscribed sphere. The probability that <math>P</math> lies inside one of the five small spheres is closest to <br />
<br />
<math> \mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4 </math><br />
<br />
[[1999 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
The number of ordered pairs of integers <math>(m,n)</math> for which <math>mn \ge 0</math> and<br />
<br />
<cmath>m^3 + n^3 + 99mn = 33^3</cmath><br />
<br />
is equal to<br />
<br />
<math> \mathrm{(A) \ }2 \qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ }35 \qquad \mathrm{(E) \ } 99</math><br />
<br />
[[1999 AHSME Problems/Problem 30|Solution]]<br />
<br />
<br />
== See also ==<br />
*[[AHSME]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1998_AHSME_Problems/Problem_29&diff=447781998 AHSME Problems/Problem 292012-02-16T02:59:23Z<p>Freddylukai: /* Problem */</p>
<hr />
<div>== Problem==<br />
A point <math>(x,y)</math> in the plane is called a lattice point if both <math>x</math> and <math>y</math> are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to<br />
<br />
<math> \mathrm{(A) \ } 4.0 \qquad \mathrm{(B) \ } 4.2 \qquad \mathrm{(C) \ } 4.5 \qquad \mathrm{(D) \ } 5.0 \qquad \mathrm{(E) \ } 5.6</math><br />
<br />
<br />
== Solution ==<br />
Sadly, I don't actually have a solution. However, after doing some work on Geogebra, I have convinced myself that the answer is almost certainly A<br />
<br />
== See also ==<br />
{{AHSME box|year=1998|num-b=28|num-a=30}}</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1996_AHSME_Problems&diff=446631996 AHSME Problems2012-02-11T03:19:20Z<p>Freddylukai: /* Problem 11 */</p>
<hr />
<div>==Problem 1==<br />
<br />
The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?<br />
<br />
<math> \begin{tabular}{r}&\ \texttt{6 4 1}\\ \texttt{8 5 2} &+\texttt{9 7 3}\\ \hline \texttt{2 4 5 6}\end{tabular} </math><br />
<br />
<br />
<math> \text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E)}\ 8 </math><br />
<br />
<br />
[[1996 AHSME Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
<br />
Each day Walter gets <math>3</math> dollars for doing his chores or <math>5</math> dollars for doing them exceptionally well. After <math>10</math> days of doing his chores daily, Walter has received a total of <math>36</math> dollars. On how many days did Walter do them exceptionally well?<br />
<br />
<math> \text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 7 </math><br />
<br />
[[1996 AHSME Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
<br />
<math> \frac{(3!)!}{3!}= </math><br />
<br />
<math> \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 6\qquad\text{(D)}\ 40\qquad\text{(E)}\ 120 </math><br />
<br />
[[1996 AHSME Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
<br />
Six numbers from a list of nine integers are <math>7,8,3,5, 9</math> and <math>5</math>. The largest possible value of the median of all nine numbers in this list is<br />
<br />
<math> \text{(A)}\ 5\qquad\text{(B)}\6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 </math><br />
<br />
[[1996 AHSME Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
<br />
Given that <math> 0 < a < b < c < d </math>, which of the following is the largest?<br />
<br />
<math> \text{(A)}\ \frac{a+b}{c+d} \qquad\text{(B)}\ \frac{a+d}{b+c} \qquad\text{(C)}\ \frac{b+c}{a+d} \qquad\text{(D)}\ \frac{b+d}{a+c} \qquad\text{(E)}\ \frac{c+d}{a+b} </math><br />
<br />
[[1996 AHSME Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
<br />
If <math> f(x) = x^{(x+1)}(x+2)^{(x+3)} </math>, then <math> f(0)+f(-1)+f(-2)+f(-3) = </math><br />
<br />
<math> \text{(A)}\ -\frac{8}{9}\qquad\text{(B)}\ 0\qquad\text{(C)}\ \frac{8}{9}\qquad\text{(D)}\ 1\qquad\text{(E)}\ \frac{10}{9} </math><br />
<br />
[[1996 AHSME Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
<br />
A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges <math>4.95</math> for the father and <math>0.45</math> for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is <math>9.45</math>, which of the following could be the age of the youngest child? <br />
<br />
<math> \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5 </math><br />
<br />
[[1996 AHSME Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
<br />
If <math>3 = k\cdot 2^r</math> and <math>15 = k\cdot 4^r</math>, then <math>r = </math><br />
<br />
<math> \text{(A)}\ -\log_{2}5\qquad\text{(B)}\ \log_{5}2\qquad\text{(C)}\ \log_{10}5\qquad\text{(D)}\ \log_{2}5\qquad\text{(E)}\ \frac{5}{2} </math><br />
<br />
[[1996 AHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
<br />
Triangle <math>PAB</math> and square <math>ABCD</math> are in perpendicular planes. Given that <math>PA = 3, PB = 4</math> and <math>AB = 5</math>, what is <math>PD</math>? <br />
<asy><br />
real r=sqrt(2)/2;<br />
draw(origin--(8,0)--(8,-1)--(0,-1)--cycle);<br />
draw(origin--(8,0)--(8+r, r)--(r,r)--cycle);<br />
filldraw(origin--(-6*r, -6*r)--(8-6*r, -6*r)--(8, 0)--cycle, white, black);<br />
filldraw(origin--(8,0)--(8,6)--(0,6)--cycle, white, black);<br />
pair A=(6,0), B=(2,0), C=(2,4), D=(6,4), P=B+1*dir(-65);<br />
draw(A--P--B--C--D--cycle);<br />
dot(A^^B^^C^^D^^P);<br />
label("$A$", A, dir((4,2)--A));<br />
label("$B$", B, dir((4,2)--B));<br />
label("$C$", C, dir((4,2)--C));<br />
label("$D$", D, dir((4,2)--D));<br />
label("$P$", P, dir((4,2)--P));</asy><br />
<br />
<math> \text{(A)}\ 5\qquad\text{(B)}\ \sqrt{34} \qquad\text{(C)}\ \sqrt{41}\qquad\text{(D)}\ 2\sqrt{13}\qquad\text{(E)}\ 8 </math><br />
<br />
[[1996 AHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
<br />
How many line segments have both their endpoints located at the vertices of a given cube?<br />
<br />
<math> \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 24\qquad\text{(D)}\ 28\qquad\text{(E)}\ 56 </math><br />
<br />
[[1996 AHSME Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
<br />
Given a circle of radius <math>2</math>, there are many line segments of length <math>2</math> that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.<br />
<br />
<math> \text{(A)}\ \frac{\pi} 4\qquad\text{(B)}\ 4-\pi\qquad\text{(C)}\ \frac{\pi} 2\qquad\text{(D)}\ \pi\qquad\text{(E)}\ 2\pi </math><br />
<br />
[[1996 AHSME Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A function <math>f</math> from the integers to the integers is defined as follows: <br />
<br />
<cmath> f(x) =\begin{cases}n+3 &\text{if n is odd}\\ n/2 &\text{if n is even}\end{cases} </cmath><br />
<br />
Suppose <math>k</math> is odd and <math>f(f(f(k))) = 27</math>. What is the sum of the digits of <math>k</math>?<br />
<br />
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15 </math><br />
<br />
<br />
[[1996 AHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Sunny runs at a steady rate, and Moonbeam runs <math>m</math> times as fast, where <math>m</math> is a number greater than 1. If Moonbeam gives Sunny a head start of <math>h</math> meters, how many meters must Moonbeam run to overtake Sunny? <br />
<br />
<math> \text{(A)}\ hm\qquad\text{(B)}\ \frac{h}{h+m}\qquad\text{(C)}\ \frac{h}{m-1}\qquad\text{(D)}\ \frac{hm}{m-1}\qquad\text{(E)}\ \frac{h+m}{m-1} </math><br />
<br />
[[1996 AHSME Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
Let <math>E(n)</math> denote the sum of the even digits of <math>n</math>. For example, <math> E(5681) = 6+8 = 14 </math>. Find <math> E(1)+E(2)+E(3)+\cdots+E(100) </math><br />
<br />
<math> \text{(A)}\ 200\qquad\text{(B)}\ 360\qquad\text{(C)}\ 400\qquad\text{(D)}\ 900\qquad\text{(E)}\ 2250 </math><br />
<br />
[[1996 AHSME Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
Two opposite sides of a rectangle are each divided into <math>n</math> congruent segments, and the endpoints of one segment are joined to the center to form triangle <math>A</math>. The other sides are each divided into <math>m</math> congruent segments, and the endpoints of one of these segments are joined to the center to form triangle <math>B</math>. [See figure for <math>n=5, m=7</math>.] What is the ratio of the area of triangle <math>A</math> to the area of triangle <math>B</math>?<br />
<br />
<asy><br />
int i;<br />
for(i=0; i<8; i=i+1) {<br />
dot((i,0)^^(i,5));<br />
}<br />
for(i=1; i<5; i=i+1) {<br />
dot((0,i)^^(7,i));<br />
}<br />
draw(origin--(7,0)--(7,5)--(0,5)--cycle, linewidth(0.8));<br />
pair P=(3.5, 2.5);<br />
draw((0,4)--P--(0,3)^^(2,0)--P--(3,0));<br />
label("$B$", (2.3,0), NE);<br />
label("$A$", (0,3.7), SE);<br />
</asy><br />
<br />
<math> \text{(A)}\ 1\qquad\text{(B)}\ m/n\qquad\text{(C)}\ n/m\qquad\text{(D)}\ 2m/n\qquad\text{(E)}\ 2n/m </math><br />
<br />
[[1996 AHSME Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one "2" is tossed?<br />
<br />
<math> \text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{91}{216}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{8}{15}\qquad\text{(E)}\ \frac{7}{12} </math><br />
<br />
[[1996 AHSME Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
In rectangle <math>ABCD</math>, angle <math>C</math> is trisected by <math>\overline{CF}</math> and <math>\overline{CE}</math>, where <math>E</math> is on <math>\overline{AB}</math>, <math>F</math> is on <math>\overline{AD}</math>, <math>BE=6</math> and <math>AF=2</math>. Which of the following is closest to the area of the rectangle <math>ABCD</math>?<br />
<asy><br />
pair A=origin, B=(10,0), C=(10,7), D=(0,7), E=(5,0), F=(0,2);<br />
draw(A--B--C--D--cycle, linewidth(0.8));<br />
draw(E--C--F);<br />
dot(A^^B^^C^^D^^E^^F);<br />
label("$A$", A, dir((5, 3.5)--A));<br />
label("$B$", B, dir((5, 3.5)--B));<br />
label("$C$", C, dir((5, 3.5)--C));<br />
label("$D$", D, dir((5, 3.5)--D));<br />
label("$E$", E, dir((5, 3.5)--E));<br />
label("$F$", F, dir((5, 3.5)--F));<br />
label("$2$", (0,1), dir(0));<br />
label("$6$", (7.5,0), N);</asy><br />
<math> \text{(A)}\ 110\qquad\text{(B)}\ 120\qquad\text{(C)}\ 130\qquad\text{(D)}\ 140\qquad\text{(E)}\ 150 </math><br />
<br />
[[1996 AHSME Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A circle of radius <math>2</math> has center at <math>(2,0)</math>. A circle of radius <math>1</math> has center at <math>(5,0)</math>. A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the <math>y</math>-intercept of the line?<br />
<br />
<math> \text{(A)}\ \sqrt{2}/4\qquad\text{(B)}\ 8/3\qquad\text{(C)}\ 1+\sqrt 3\qquad\text{(D)}\ 2\sqrt 2\qquad\text{(E)}\ 3 </math><br />
<br />
[[1996 AHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
The midpoints of the sides of a regular hexagon <math>ABCDEF</math> are joined to form a smaller hexagon. What fraction of the area of <math>ABCDEF</math> is enclosed by the smaller hexagon? <br />
<br />
<asy><br />
size(120);<br />
draw(rotate(30)*polygon(6));<br />
draw(scale(2/sqrt(3))*polygon(6));<br />
pair A=2/sqrt(3)*dir(120), B=2/sqrt(3)*dir(180), C=2/sqrt(3)*dir(240), D=2/sqrt(3)*dir(300), E=2/sqrt(3)*dir(0), F=2/sqrt(3)*dir(60);<br />
dot(A^^B^^C^^D^^E^^F);<br />
label("$A$", A, dir(origin--A));<br />
label("$B$", B, dir(origin--B));<br />
label("$C$", C, dir(origin--C));<br />
label("$D$", D, dir(origin--D));<br />
label("$E$", E, dir(origin--E));<br />
label("$F$", F, dir(origin--F));<br />
</asy><br />
<br />
<math> \text{(A)}\ \frac{1}{2}\qquad\text{(B)}\ \frac{\sqrt 3}{3}\qquad\text{(C)}\ \frac{2}{3}\qquad\text{(D)}\ \frac{3}{4}\qquad\text{(E)}\ \frac{\sqrt 3}{2} </math><br />
<br />
[[1996 AHSME Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
In the xy-plane, what is the length of the shortest path from <math>(0,0)</math> to <math>(12,16)</math> that does not go inside the circle <math> (x-6)^{2}+(y-8)^{2}= 25 </math>?<br />
<br />
<math> \text{(A)}\ 10\sqrt 3\qquad\text{(B)}\ 10\sqrt 5\qquad\text{(C)}\ 10\sqrt 3+\frac{ 5\pi}{3}\qquad\text{(D)}\ 40\frac{\sqrt{3}}{3}\qquad\text{(E)}\ 10+5\pi </math><br />
<br />
[[1996 AHSME Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
Triangles <math>ABC</math> and <math>ABD</math> are isosceles with <math>AB=AC=BD</math>, and <math>BD</math> intersects <math>AC</math> at <math>E</math>. If <math>BD</math> is perpendicular to <math>AC</math>, then <math> \angle C+\angle D </math> is<br />
<br />
<asy><br />
size(120);<br />
pair B=origin, A=1*dir(70), M=foot(A, B, (3,0)), C=reflect(A, M)*B, E=foot(B, A, C), D=1*dir(20);<br />
dot(A^^B^^C^^D^^E);<br />
draw(A--D--B--A--C--B);<br />
markscalefactor=0.005;<br />
draw(rightanglemark(A, E, B));<br />
dot(A^^B^^C^^D^^E);<br />
pair point=midpoint(A--M);<br />
label("$A$", A, dir(point--A));<br />
label("$B$", B, dir(point--B));<br />
label("$C$", C, dir(point--C));<br />
label("$D$", D, dir(point--D));<br />
label("$E$", E, dir(point--E));<br />
</asy><br />
<br />
<math> \text{(A)}\ 115^\circ\qquad\text{(B)}\ 120^\circ\qquad\text{(C)}\ 130^\circ\qquad\text{(D)}\ 135^\circ\qquad\text{(E)}\ \text{not uniquely determined} </math><br />
<br />
[[1996 AHSME Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
Four distinct points, <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, are to be selected from <math>1996</math> points <br />
evenly spaced around a circle. All quadruples are equally likely to be chosen. <br />
What is the probability that the chord <math>AB</math> intersects the chord <math>CD</math>? <br />
<br />
<math> \text{(A)}\ \frac{1}{4}\qquad\text{(B)}\ \frac{1}{3}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{2}{3}\qquad\text{(E)}\ \frac{3}{4} </math><br />
<br />
[[1996 AHSME Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
The sum of the lengths of the twelve edges of a rectangular box is <math>140</math>, and <br />
the distance from one corner of the box to the farthest corner is <math>21</math>. The total <br />
surface area of the box is <br />
<br />
<math> \text{(A)}\ 776\qquad\text{(B)}\ 784\qquad\text{(C)}\ 798\qquad\text{(D)}\ 800\qquad\text{(E)}\ 812 </math><br />
<br />
[[1996 AHSME Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
The sequence <math> 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots </math> consists of <math>1</math>’s separated by blocks of <math>2</math>’s with <math>n</math> <math>2</math>’s in the <math>n^{th}</math> block. The sum of the first <math>1234</math> terms of this sequence is<br />
<br />
<math> \text{(A)}\ 1996\qquad\text{(B)}\ 2419\qquad\text{(C)}\ 2429\qquad\text{(D)}\ 2439\qquad\text{(E)}\ 2449 </math><br />
<br />
[[1996 AHSME Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
Given that <math>x^2 + y^2 = 14x + 6y + 6</math>, what is the largest possible value that <math>3x + 4y</math> can have? <br />
<br />
<math> \text{(A)}\ 72\qquad\text{(B)}\ 73\qquad\text{(C)}\ 74\qquad\text{(D)}\ 75\qquad\text{(E)}\ 76 </math><br />
<br />
[[1996 AHSME Problems/Problem 25|Solution]]<br />
<br />
==Problem 26==<br />
<br />
An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely: <br />
<br />
(a) the selection of four red marbles; <br />
<br />
(b) the selection of one white and three red marbles; <br />
<br />
(c) the selection of one white, one blue, and two red marbles; and <br />
<br />
(d) the selection of one marble of each color. <br />
<br />
What is the smallest number of marbles satisfying the given condition? <br />
<br />
<math> \text{(A)}\ 19\qquad\text{(B)}\ 21\qquad\text{(C)}\ 46\qquad\text{(D)}\ 69\qquad\text{(E)}\ \text{more than 69} </math><br />
<br />
[[1996 AHSME Problems/Problem 26|Solution]]<br />
<br />
==Problem 27==<br />
<br />
Consider two solid spherical balls, one centered at <math> (0, 0,\frac{21}{2}) </math> with radius <math>6</math>, and the other centered at <math> (0, 0, 1) </math> with radius <math>\frac{9}{2}</math>. How many points with only integer coordinates (lattice points) are there in the intersection of the balls? <br />
<br />
<math> \text{(A)}\ 7\qquad\text{(B)}\ 9\qquad\text{(C)}\ 11\qquad\text{(D)}\ 13\qquad\text{(E)}\ 15 </math><br />
<br />
[[1996 AHSME Problems/Problem 27|Solution]]<br />
<br />
==Problem 28==<br />
<br />
On a <math> 4\times 4\times 3 </math> rectangular parallelepiped, vertices <math>A</math>, <math>B</math>, and <math>C</math> are adjacent to vertex <math>D</math>. The perpendicular distance from <math>D</math> to the plane containing<br />
<math>A</math>, <math>B</math>, and <math>C</math> is closest to <br />
<br />
<asy><br />
size(120);<br />
import three;<br />
currentprojection=orthographic(1, 4/5, 1/3);<br />
draw(box(O, (4,4,3)));<br />
triple A=(0,4,3), B=(0,0,0) , C=(4,4,0), D=(0,4,0);<br />
draw(A--B--C--cycle, linewidth(0.9));<br />
label("$A$", A, NE);<br />
label("$B$", B, NW);<br />
label("$C$", C, S);<br />
label("$D$", D, E);<br />
label("$4$", (4,2,0), SW);<br />
label("$4$", (2,4,0), SE);<br />
label("$3$", (0, 4, 1.5), E);<br />
</asy><br />
<br />
<math> \text{(A)}\ 1.6\qquad\text{(B)}\ 1.9\qquad\text{(C)}\ 2.1\qquad\text{(D)}\ 2.7\qquad\text{(E)}\ 2.9 </math><br />
<br />
[[1996 AHSME Problems/Problem 28|Solution]]<br />
<br />
==Problem 29==<br />
<br />
If <math>n</math> is a positive integer such that <math>2n</math> has <math>28</math> positive divisors and <math>3n</math> has <math>30</math> positive divisors, then how many positive divisors does <math>6n</math> have?<br />
<br />
<math> \text{(A)}\ 32\qquad\text{(B)}\ 34\qquad\text{(C)}\ 35\qquad\text{(D)}\ 36\qquad\text{(E)}\ 38 </math><br />
<br />
[[1996 AHSME Problems/Problem 29|Solution]]<br />
<br />
==Problem 30==<br />
<br />
A hexagon inscribed in a circle has three consecutive sides each of length <math>3</math> and three consecutive sides each of length <math>5</math>. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length <math>3</math> and the other with three sides each of length <math>5</math>, has length equal to <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.<br />
<br />
<math> \text{(A)}\ 309\qquad\text{(B)}\ 349\qquad\text{(C)}\ 369\qquad\text{(D)}\ 389\qquad\text{(E)}\ 409 </math><br />
<br />
[[1996 AHSME Problems/Problem 30|Solution]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems&diff=446511995 AHSME Problems2012-02-10T02:54:29Z<p>Freddylukai: /* Problem 19 */</p>
<hr />
<div>== Problem 1 ==<br />
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <br />
<br />
<math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qquad \mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by 3} } \qquad \mathrm{(E) \ \text{increase by 4} } </math><br />
<br />
[[1995 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math><br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \sqrt{7} } \qquad \mathrm{(C) \ 7 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 121 } </math><br />
<br />
[[1995 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How many cents are saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
If <math>M</math> is <math>30 \%</math> of <math>Q</math>, <math>Q</math> is <math>20 \%</math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math><br />
<br />
<math> \mathrm{(A) \ \frac {3}{250} } \qquad \mathrm{(B) \ \frac {3}{25} } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ \frac {6}{5} } \qquad \mathrm{(E) \ \frac {4}{3} } </math><br />
<br />
[[1995 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is <br />
<br />
<math> \mathrm{(A) \ \text{500 thousand} } \qquad \mathrm{(B) \ \text{5 million} } \qquad \mathrm{(C) \ \text{50 million} } \qquad \mathrm{(D) \ \text{500 million} } \qquad \mathrm{(E) \ \text{5 billion} } </math><br />
<br />
[[1995 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked <math>x</math>? <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);<br />
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);<br />
draw(shift(1,0)*p, dashed);<br />
label("$x$", (0.3,0.5), E);<br />
label("$A$", (1.3,0.5), E);<br />
label("$B$", (1.3,1.5), E);<br />
label("$C$", (2.3,1.5), E);<br />
label("$D$", (2.3,2.5), E);<br />
label("$E$", (3.3,2.5), E);</asy><br />
<br />
<math> \mathrm{(A) \ A } \qquad \mathrm{(B) \ B } \qquad \mathrm{(C) \ C } \qquad \mathrm{(D) \ D } \qquad \mathrm{(E) \ E } </math><br />
<br />
[[1995 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: <br />
<br />
<math> \mathrm{(A) \ 8 } \qquad \mathrm{(B) \ 25 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 75 } \qquad \mathrm{(E) \ 100 } </math><br />
<br />
[[1995 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
In <math>\triangle ABC</math>, <math>\angle C = 90^\circ, AC = 6</math> and <math>BC = 8</math>. Points <math>D</math> and <math>E</math> are on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, and <math>\angle BED = 90^\circ</math>. If <math>DE = 4</math>, then <math>BD =</math><br />
<asy> <br />
size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16)); <br />
</asy><br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ \frac {16}{3} } \qquad \mathrm{(C) \ \frac {20}{3} } \qquad \mathrm{(D) \ \frac {15}{2} } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is <br />
<asy><br />
size(100); defaultpen(linewidth(0.7)); draw(unitsquare^^(0,0)--(1,1)^^(0,1)--(1,0)^^(.5,0)--(.5,1)^^(0,.5)--(1,.5));<br />
</asy><br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 12 } \qquad \mathrm{(C) \ 14 } \qquad \mathrm{(D) \ 16 } \qquad \mathrm{(E) \ 18 } </math><br />
<br />
[[1995 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The area of the triangle bounded by the lines <math>y = x, y = - x</math> and <math>y = 6</math> is<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 12\sqrt{2} } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 24\sqrt{2} } \qquad \mathrm{(E) \ 36 } </math><br />
<br />
[[1995 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
How many base 10 four-digit numbers, <math>N = \underline{a} \underline{b} \underline{c} \underline{d}</math>, satisfy all three of the following conditions?<br />
<br />
<math>\text{(i)}</math> <math>4,000 \leq N < 6,000;</math> <br />
<br />
<math>\text{(ii)}</math> <math>N</math> <math>\text{is a multiple of 5}</math>; <br />
<br />
<math>\text{(iii)}</math> <math>3 \leq b < c \leq 6</math>.<br />
<br />
<br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 18 } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 36 } \qquad \mathrm{(E) \ 48 } </math><br />
<br />
[[1995 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
Let <math>f</math> be a linear function with the properties that <math>f(1) \leq f(2), f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true?<br />
<br />
<math> \mathrm{(A) \ f(0) < 0 } \qquad \mathrm{(B) \ f(0) = 0 } \qquad \mathrm{(C) \ f(1) < f(0) < f( - 1) } \qquad \mathrm{(D) \ f(0) = 5 } \qquad \mathrm{(E) \ f(0) > 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit <math>e</math>. Find the sum of <math>d</math> and <math>e</math>.<br />
<br />
<math>\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\<br />
+ & 8 & 2 & 9 & 4 & 3 & 0 \\<br />
\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}</math><br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ \text{more than 10} } </math><br />
<br />
[[1995 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
If <math>f(x) = ax^4 - bx^2 + x + 5</math> and <math>f( - 3) = 2</math>, then <math>f(3) =</math><br />
<br />
<math> \mathrm{(A) \ -5 } \qquad \mathrm{(B) \ -2 } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point <br />
<asy><br />
size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle);<br />
for(int i = 0; i < 5; ++i) { pair P = dir(90-i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }<br />
</asy><br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:<br />
<br />
i. The actual attendance in Atlanta is within <math>10 \%</math> of Anita's estimate.<br />
ii. Bob's estimate is within <math>10 \%</math> of the actual attendance in Boston.<br />
<br />
To the nearest 1,000, the largest possible difference between the numbers attending the two games is<br />
<br />
<math> \mathrm{(A) \ 10000 } \qquad \mathrm{(B) \ 11000 } \qquad \mathrm{(C) \ 20000 } \qquad \mathrm{(D) \ 21000 } \qquad \mathrm{(E) \ 22000 } </math><br />
<br />
[[1995 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is<br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(rotate(18)*polygon(5));<br />
real x=0.6180339887;<br />
draw(Circle((-x,0), 1));<br />
int i;<br />
for(i=0; i<5; i=i+1) {<br />
dot(origin+1*dir(36+72*i));<br />
}<br />
label("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0)));<br />
label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72)));<br />
label("$E$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144)));<br />
label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3)));<br />
label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));</asy><br />
<br />
<math> \mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 } </math><br />
<br />
[[1995 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
Two rays with common endpoint <math>O</math> forms a <math>30^\circ</math> angle. Point <math>A</math> lies on one ray, point <math>B</math> on the other ray, and <math>AB = 1</math>. The maximum possible length of <math>OB</math> is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \frac {1 + \sqrt {3}}{\sqrt 2} } \qquad \mathrm{(C) \ \sqrt{3} } \qquad \mathrm{(D) \ 2 } \qquad \mathrm{(E) \ \frac{4}{\sqrt{3}} } </math><br />
<br />
[[1995 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
Equilateral triangle <math>DEF</math> is inscribed in equilateral triangle <math>ABC</math> such that <math>\overline{DE} \perp \overline{BC}</math>. The ratio of the area of <math>\triangle DEF</math> to the area of <math>\triangle ABC</math> is<br />
<asy><br />
pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);<br />
pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3;<br />
D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));<br />
</asy><br />
<math> \mathrm{(A) \ \frac {1}{6} } \qquad \mathrm{(B) \ \frac {1}{4} } \qquad \mathrm{(C) \ \frac {1}{3} } \qquad \mathrm{(D) \ \frac {2}{5} } \qquad \mathrm{(E) \ \frac {1}{2} } </math><br />
<br />
[[1995 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
If <math>a,b</math> and <math>c</math> are three (not necessarily different) numbers chosen randomly and with replacement from the set <math>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is<br />
<br />
<math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } </math><br />
<br />
[[1995 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
Two nonadjacent vertices of a rectangle are <math>(4,3)</math> and <math>(-4,-3)</math>, and the coordinates of the other two vertices are integers. The number of such rectangles is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is<br />
<br />
<math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 } </math><br />
<br />
[[1995 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The sides of a triangle have lengths 11,15, and <math>k</math>, where <math>k</math> is an integer. For how many values of <math>k</math> is the triangle obtuse?<br />
<br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 } </math><br />
<br />
[[1995 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that<br />
<br />
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath><br />
<br />
What is <math>A + B + C</math>?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list? <br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } </math><br />
<br />
[[1995 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(Circle(origin, 5));<br />
pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D);<br />
draw(A--B^^C--D--F);<br />
dot(O^^A^^B^^C^^D^^E^^F);<br />
markscalefactor=0.05;<br />
draw(rightanglemark(B, O, D));<br />
label("$A$", A, dir(O--A));<br />
label("$B$", B, dir(O--B));<br />
label("$C$", C, dir(O--C));<br />
label("$D$", D, dir(O--D));<br />
label("$F$", F, dir(O--F));<br />
label("$O$", O, NW);<br />
label("$E$", E, SE);</asy><br />
<br />
<math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math><br />
<br />
[[1995 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.<br />
<br />
<cmath>\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\<br />
& & & & 1 & & 1 & & & & \\<br />
& & & 2 & & 2 & & 2 & & & \\<br />
& & 3 & & 4 & & 4 & & 3 & & \\<br />
& 4 & & 7 & & 8 & & 7 & & 4 & \\<br />
5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular}</cmath><br />
<br />
Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } </math><br />
<br />
[[1995 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is<br />
<asy><br />
// note: diagram deliberately not to scale -- azjps<br />
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }<br />
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
real min = -0.6, step = 0.5;<br />
pair[] A, B; D(unitcircle);<br />
for(int i = 0; i < 3; ++i) {<br />
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);<br />
D(D(A[i])--D(B[i]));<br />
}<br />
MP("10",(A[0]+B[0])/2,N);<br />
MP("\sqrt{a}",(A[1]+B[1])/2,N);<br />
MP("14",(A[2]+B[2])/2,N);<br />
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);<br />
</asy><br />
<math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } </math><br />
<br />
[[1995 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?<br />
<br />
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 } </math><br />
<br />
[[1995 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is<br />
<asy><br />
size(150); defaultpen(linewidth(0.7)); pair slant = (2,1); <br />
for(int i = 0; i < 4; ++i) <br />
draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); <br />
for(int i = 1; i < 4; ++i)<br />
draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);<br />
</asy><br />
<math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } </math><br />
<br />
[[1995 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
{{AHSME box|year=1995|before=[[1994 AHSME Problems|1994 AHSME]]|after=[[1996 AHSME Problems|1996 AHSME]]}}<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems/Problem_15&diff=446501995 AHSME Problems/Problem 152012-02-10T02:49:58Z<p>Freddylukai: /* Solution */ redid incorrect solution</p>
<hr />
<div>==Problem==<br />
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point <br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
==Solution==<br />
Let us see how the bug moves.<br />
<br />
First, we see that if it starts at point 5, it moves to point 1.<br />
<br />
At point 1, it moves to point 2.<br />
<br />
At point 2, since 2 is even, it moves to point 4.<br />
<br />
Then at point 4, it moves to point 1.<br />
<br />
We can see that at this point, the bug will cycle between <math>\{{1,2,4}\}%</math><br />
<br />
More specifically, we can see that all numbers congruent to 0 (mod 3) will have the bug on point 2 on that step number.<br />
<br />
Thus, we can conclude that the answer is <math>\fbox{\text{(B)}}</math><br />
<br />
==See also==<br />
{{AHSME box|year=1995|num-b=14|num-a=16}} <br />
<br />
[[Category:Introductory Number Theory Problems]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems&diff=446491995 AHSME Problems2012-02-10T02:26:16Z<p>Freddylukai: /* Problem 11 */</p>
<hr />
<div>== Problem 1 ==<br />
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <br />
<br />
<math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qquad \mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by 3} } \qquad \mathrm{(E) \ \text{increase by 4} } </math><br />
<br />
[[1995 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math><br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \sqrt{7} } \qquad \mathrm{(C) \ 7 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 121 } </math><br />
<br />
[[1995 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How many cents are saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
If <math>M</math> is <math>30 \%</math> of <math>Q</math>, <math>Q</math> is <math>20 \%</math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math><br />
<br />
<math> \mathrm{(A) \ \frac {3}{250} } \qquad \mathrm{(B) \ \frac {3}{25} } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ \frac {6}{5} } \qquad \mathrm{(E) \ \frac {4}{3} } </math><br />
<br />
[[1995 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is <br />
<br />
<math> \mathrm{(A) \ \text{500 thousand} } \qquad \mathrm{(B) \ \text{5 million} } \qquad \mathrm{(C) \ \text{50 million} } \qquad \mathrm{(D) \ \text{500 million} } \qquad \mathrm{(E) \ \text{5 billion} } </math><br />
<br />
[[1995 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked <math>x</math>? <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);<br />
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);<br />
draw(shift(1,0)*p, dashed);<br />
label("$x$", (0.3,0.5), E);<br />
label("$A$", (1.3,0.5), E);<br />
label("$B$", (1.3,1.5), E);<br />
label("$C$", (2.3,1.5), E);<br />
label("$D$", (2.3,2.5), E);<br />
label("$E$", (3.3,2.5), E);</asy><br />
<br />
<math> \mathrm{(A) \ A } \qquad \mathrm{(B) \ B } \qquad \mathrm{(C) \ C } \qquad \mathrm{(D) \ D } \qquad \mathrm{(E) \ E } </math><br />
<br />
[[1995 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: <br />
<br />
<math> \mathrm{(A) \ 8 } \qquad \mathrm{(B) \ 25 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 75 } \qquad \mathrm{(E) \ 100 } </math><br />
<br />
[[1995 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
In <math>\triangle ABC</math>, <math>\angle C = 90^\circ, AC = 6</math> and <math>BC = 8</math>. Points <math>D</math> and <math>E</math> are on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, and <math>\angle BED = 90^\circ</math>. If <math>DE = 4</math>, then <math>BD =</math><br />
<asy> <br />
size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16)); <br />
</asy><br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ \frac {16}{3} } \qquad \mathrm{(C) \ \frac {20}{3} } \qquad \mathrm{(D) \ \frac {15}{2} } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is <br />
<asy><br />
size(100); defaultpen(linewidth(0.7)); draw(unitsquare^^(0,0)--(1,1)^^(0,1)--(1,0)^^(.5,0)--(.5,1)^^(0,.5)--(1,.5));<br />
</asy><br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 12 } \qquad \mathrm{(C) \ 14 } \qquad \mathrm{(D) \ 16 } \qquad \mathrm{(E) \ 18 } </math><br />
<br />
[[1995 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The area of the triangle bounded by the lines <math>y = x, y = - x</math> and <math>y = 6</math> is<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 12\sqrt{2} } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 24\sqrt{2} } \qquad \mathrm{(E) \ 36 } </math><br />
<br />
[[1995 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
How many base 10 four-digit numbers, <math>N = \underline{a} \underline{b} \underline{c} \underline{d}</math>, satisfy all three of the following conditions?<br />
<br />
<math>\text{(i)}</math> <math>4,000 \leq N < 6,000;</math> <br />
<br />
<math>\text{(ii)}</math> <math>N</math> <math>\text{is a multiple of 5}</math>; <br />
<br />
<math>\text{(iii)}</math> <math>3 \leq b < c \leq 6</math>.<br />
<br />
<br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 18 } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 36 } \qquad \mathrm{(E) \ 48 } </math><br />
<br />
[[1995 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
Let <math>f</math> be a linear function with the properties that <math>f(1) \leq f(2), f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true?<br />
<br />
<math> \mathrm{(A) \ f(0) < 0 } \qquad \mathrm{(B) \ f(0) = 0 } \qquad \mathrm{(C) \ f(1) < f(0) < f( - 1) } \qquad \mathrm{(D) \ f(0) = 5 } \qquad \mathrm{(E) \ f(0) > 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit <math>e</math>. Find the sum of <math>d</math> and <math>e</math>.<br />
<br />
<math>\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\<br />
+ & 8 & 2 & 9 & 4 & 3 & 0 \\<br />
\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}</math><br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ \text{more than 10} } </math><br />
<br />
[[1995 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
If <math>f(x) = ax^4 - bx^2 + x + 5</math> and <math>f( - 3) = 2</math>, then <math>f(3) =</math><br />
<br />
<math> \mathrm{(A) \ -5 } \qquad \mathrm{(B) \ -2 } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point <br />
<asy><br />
size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle);<br />
for(int i = 0; i < 5; ++i) { pair P = dir(90-i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }<br />
</asy><br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:<br />
<br />
i. The actual attendance in Atlanta is within <math>10 \%</math> of Anita's estimate.<br />
ii. Bob's estimate is within <math>10 \%</math> of the actual attendance in Boston.<br />
<br />
To the nearest 1,000, the largest possible difference between the numbers attending the two games is<br />
<br />
<math> \mathrm{(A) \ 10000 } \qquad \mathrm{(B) \ 11000 } \qquad \mathrm{(C) \ 20000 } \qquad \mathrm{(D) \ 21000 } \qquad \mathrm{(E) \ 22000 } </math><br />
<br />
[[1995 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is<br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(rotate(18)*polygon(5));<br />
real x=0.6180339887;<br />
draw(Circle((-x,0), 1));<br />
int i;<br />
for(i=0; i<5; i=i+1) {<br />
dot(origin+1*dir(36+72*i));<br />
}<br />
label("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0)));<br />
label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72)));<br />
label("$E$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144)));<br />
label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3)));<br />
label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));</asy><br />
<br />
<math> \mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 } </math><br />
<br />
[[1995 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
Two rays with common endpoint <math>O</math> forms a <math>30^\circ</math> angle. Point <math>A</math> lies on one ray, point <math>B</math> on the other ray, and <math>AB = 1</math>. The maximum possible length of <math>OB</math> is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \frac {1 + \sqrt {3}}{\sqrt 2} } \qquad \mathrm{(C) \ \sqrt{3} } \qquad \mathrm{(D) \ 2 } \qquad \mathrm{(E) \ \frac{4}{\sqrt{3}} } </math><br />
<br />
[[1995 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
Equilateral triangle <math>DEF</math> is inscribed in equilateral triangle <math>ABC</math> such that <math>\overline{DE} \perp \overline{BC}</math>. The reatio of the area of <math>\triangle DEF</math> to the area of <math>\triangle ABC</math> is<br />
<asy><br />
pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);<br />
pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3;<br />
D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));<br />
</asy><br />
<math> \mathrm{(A) \ \frac {1}{6} } \qquad \mathrm{(B) \ \frac {1}{4} } \qquad \mathrm{(C) \ \frac {1}{3} } \qquad \mathrm{(D) \ \frac {2}{5} } \qquad \mathrm{(E) \ \frac {1}{2} } </math><br />
<br />
[[1995 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
If <math>a,b</math> and <math>c</math> are three (not necessarily different) numbers chosen randomly and with replacement from the set <math>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is<br />
<br />
<math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } </math><br />
<br />
[[1995 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
Two nonadjacent vertices of a rectangle are <math>(4,3)</math> and <math>(-4,-3)</math>, and the coordinates of the other two vertices are integers. The number of such rectangles is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is<br />
<br />
<math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 } </math><br />
<br />
[[1995 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The sides of a triangle have lengths 11,15, and <math>k</math>, where <math>k</math> is an integer. For how many values of <math>k</math> is the triangle obtuse?<br />
<br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 } </math><br />
<br />
[[1995 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that<br />
<br />
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath><br />
<br />
What is <math>A + B + C</math>?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list? <br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } </math><br />
<br />
[[1995 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(Circle(origin, 5));<br />
pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D);<br />
draw(A--B^^C--D--F);<br />
dot(O^^A^^B^^C^^D^^E^^F);<br />
markscalefactor=0.05;<br />
draw(rightanglemark(B, O, D));<br />
label("$A$", A, dir(O--A));<br />
label("$B$", B, dir(O--B));<br />
label("$C$", C, dir(O--C));<br />
label("$D$", D, dir(O--D));<br />
label("$F$", F, dir(O--F));<br />
label("$O$", O, NW);<br />
label("$E$", E, SE);</asy><br />
<br />
<math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math><br />
<br />
[[1995 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.<br />
<br />
<cmath>\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\<br />
& & & & 1 & & 1 & & & & \\<br />
& & & 2 & & 2 & & 2 & & & \\<br />
& & 3 & & 4 & & 4 & & 3 & & \\<br />
& 4 & & 7 & & 8 & & 7 & & 4 & \\<br />
5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular}</cmath><br />
<br />
Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } </math><br />
<br />
[[1995 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is<br />
<asy><br />
// note: diagram deliberately not to scale -- azjps<br />
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }<br />
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
real min = -0.6, step = 0.5;<br />
pair[] A, B; D(unitcircle);<br />
for(int i = 0; i < 3; ++i) {<br />
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);<br />
D(D(A[i])--D(B[i]));<br />
}<br />
MP("10",(A[0]+B[0])/2,N);<br />
MP("\sqrt{a}",(A[1]+B[1])/2,N);<br />
MP("14",(A[2]+B[2])/2,N);<br />
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);<br />
</asy><br />
<math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } </math><br />
<br />
[[1995 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?<br />
<br />
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 } </math><br />
<br />
[[1995 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is<br />
<asy><br />
size(150); defaultpen(linewidth(0.7)); pair slant = (2,1); <br />
for(int i = 0; i < 4; ++i) <br />
draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); <br />
for(int i = 1; i < 4; ++i)<br />
draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);<br />
</asy><br />
<math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } </math><br />
<br />
[[1995 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
{{AHSME box|year=1995|before=[[1994 AHSME Problems|1994 AHSME]]|after=[[1996 AHSME Problems|1996 AHSME]]}}<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems&diff=446481995 AHSME Problems2012-02-10T02:21:36Z<p>Freddylukai: /* Problem 6 */</p>
<hr />
<div>== Problem 1 ==<br />
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <br />
<br />
<math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qquad \mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by 3} } \qquad \mathrm{(E) \ \text{increase by 4} } </math><br />
<br />
[[1995 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math><br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \sqrt{7} } \qquad \mathrm{(C) \ 7 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 121 } </math><br />
<br />
[[1995 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How many cents are saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
If <math>M</math> is <math>30 \%</math> of <math>Q</math>, <math>Q</math> is <math>20 \%</math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math><br />
<br />
<math> \mathrm{(A) \ \frac {3}{250} } \qquad \mathrm{(B) \ \frac {3}{25} } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ \frac {6}{5} } \qquad \mathrm{(E) \ \frac {4}{3} } </math><br />
<br />
[[1995 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is <br />
<br />
<math> \mathrm{(A) \ \text{500 thousand} } \qquad \mathrm{(B) \ \text{5 million} } \qquad \mathrm{(C) \ \text{50 million} } \qquad \mathrm{(D) \ \text{500 million} } \qquad \mathrm{(E) \ \text{5 billion} } </math><br />
<br />
[[1995 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked <math>x</math>? <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);<br />
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);<br />
draw(shift(1,0)*p, dashed);<br />
label("$x$", (0.3,0.5), E);<br />
label("$A$", (1.3,0.5), E);<br />
label("$B$", (1.3,1.5), E);<br />
label("$C$", (2.3,1.5), E);<br />
label("$D$", (2.3,2.5), E);<br />
label("$E$", (3.3,2.5), E);</asy><br />
<br />
<math> \mathrm{(A) \ A } \qquad \mathrm{(B) \ B } \qquad \mathrm{(C) \ C } \qquad \mathrm{(D) \ D } \qquad \mathrm{(E) \ E } </math><br />
<br />
[[1995 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: <br />
<br />
<math> \mathrm{(A) \ 8 } \qquad \mathrm{(B) \ 25 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 75 } \qquad \mathrm{(E) \ 100 } </math><br />
<br />
[[1995 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
In <math>\triangle ABC</math>, <math>\angle C = 90^\circ, AC = 6</math> and <math>BC = 8</math>. Points <math>D</math> and <math>E</math> are on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, and <math>\angle BED = 90^\circ</math>. If <math>DE = 4</math>, then <math>BD =</math><br />
<asy> <br />
size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16)); <br />
</asy><br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ \frac {16}{3} } \qquad \mathrm{(C) \ \frac {20}{3} } \qquad \mathrm{(D) \ \frac {15}{2} } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is <br />
<asy><br />
size(100); defaultpen(linewidth(0.7)); draw(unitsquare^^(0,0)--(1,1)^^(0,1)--(1,0)^^(.5,0)--(.5,1)^^(0,.5)--(1,.5));<br />
</asy><br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 12 } \qquad \mathrm{(C) \ 14 } \qquad \mathrm{(D) \ 16 } \qquad \mathrm{(E) \ 18 } </math><br />
<br />
[[1995 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The area of the triangle bounded by the lines <math>y = x, y = - x</math> and <math>y = 6</math> is<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 12\sqrt{2} } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 24\sqrt{2} } \qquad \mathrm{(E) \ 36 } </math><br />
<br />
[[1995 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
How many base 10 four-digit numbers, <math>N = \underline{a} \underline{b} \underline{c} \underline{d}</math>, satisfy all three of the following conditions?<br />
<br />
(i) <math>4,000 \leq N < 6,000;</math> (ii) <math>N</math> is a multiple of 5; (iii) <math>3 \leq b < c \leq 6</math>.<br />
<br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 18 } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 36 } \qquad \mathrm{(E) \ 48 } </math><br />
<br />
[[1995 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
Let <math>f</math> be a linear function with the properties that <math>f(1) \leq f(2), f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true?<br />
<br />
<math> \mathrm{(A) \ f(0) < 0 } \qquad \mathrm{(B) \ f(0) = 0 } \qquad \mathrm{(C) \ f(1) < f(0) < f( - 1) } \qquad \mathrm{(D) \ f(0) = 5 } \qquad \mathrm{(E) \ f(0) > 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit <math>e</math>. Find the sum of <math>d</math> and <math>e</math>.<br />
<br />
<math>\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\<br />
+ & 8 & 2 & 9 & 4 & 3 & 0 \\<br />
\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}</math><br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ \text{more than 10} } </math><br />
<br />
[[1995 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
If <math>f(x) = ax^4 - bx^2 + x + 5</math> and <math>f( - 3) = 2</math>, then <math>f(3) =</math><br />
<br />
<math> \mathrm{(A) \ -5 } \qquad \mathrm{(B) \ -2 } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point <br />
<asy><br />
size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle);<br />
for(int i = 0; i < 5; ++i) { pair P = dir(90-i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }<br />
</asy><br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:<br />
<br />
i. The actual attendance in Atlanta is within <math>10 \%</math> of Anita's estimate.<br />
ii. Bob's estimate is within <math>10 \%</math> of the actual attendance in Boston.<br />
<br />
To the nearest 1,000, the largest possible difference between the numbers attending the two games is<br />
<br />
<math> \mathrm{(A) \ 10000 } \qquad \mathrm{(B) \ 11000 } \qquad \mathrm{(C) \ 20000 } \qquad \mathrm{(D) \ 21000 } \qquad \mathrm{(E) \ 22000 } </math><br />
<br />
[[1995 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is<br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(rotate(18)*polygon(5));<br />
real x=0.6180339887;<br />
draw(Circle((-x,0), 1));<br />
int i;<br />
for(i=0; i<5; i=i+1) {<br />
dot(origin+1*dir(36+72*i));<br />
}<br />
label("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0)));<br />
label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72)));<br />
label("$E$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144)));<br />
label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3)));<br />
label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));</asy><br />
<br />
<math> \mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 } </math><br />
<br />
[[1995 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
Two rays with common endpoint <math>O</math> forms a <math>30^\circ</math> angle. Point <math>A</math> lies on one ray, point <math>B</math> on the other ray, and <math>AB = 1</math>. The maximum possible length of <math>OB</math> is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \frac {1 + \sqrt {3}}{\sqrt 2} } \qquad \mathrm{(C) \ \sqrt{3} } \qquad \mathrm{(D) \ 2 } \qquad \mathrm{(E) \ \frac{4}{\sqrt{3}} } </math><br />
<br />
[[1995 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
Equilateral triangle <math>DEF</math> is inscribed in equilateral triangle <math>ABC</math> such that <math>\overline{DE} \perp \overline{BC}</math>. The reatio of the area of <math>\triangle DEF</math> to the area of <math>\triangle ABC</math> is<br />
<asy><br />
pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);<br />
pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3;<br />
D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));<br />
</asy><br />
<math> \mathrm{(A) \ \frac {1}{6} } \qquad \mathrm{(B) \ \frac {1}{4} } \qquad \mathrm{(C) \ \frac {1}{3} } \qquad \mathrm{(D) \ \frac {2}{5} } \qquad \mathrm{(E) \ \frac {1}{2} } </math><br />
<br />
[[1995 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
If <math>a,b</math> and <math>c</math> are three (not necessarily different) numbers chosen randomly and with replacement from the set <math>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is<br />
<br />
<math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } </math><br />
<br />
[[1995 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
Two nonadjacent vertices of a rectangle are <math>(4,3)</math> and <math>(-4,-3)</math>, and the coordinates of the other two vertices are integers. The number of such rectangles is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is<br />
<br />
<math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 } </math><br />
<br />
[[1995 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The sides of a triangle have lengths 11,15, and <math>k</math>, where <math>k</math> is an integer. For how many values of <math>k</math> is the triangle obtuse?<br />
<br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 } </math><br />
<br />
[[1995 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that<br />
<br />
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath><br />
<br />
What is <math>A + B + C</math>?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list? <br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } </math><br />
<br />
[[1995 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(Circle(origin, 5));<br />
pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D);<br />
draw(A--B^^C--D--F);<br />
dot(O^^A^^B^^C^^D^^E^^F);<br />
markscalefactor=0.05;<br />
draw(rightanglemark(B, O, D));<br />
label("$A$", A, dir(O--A));<br />
label("$B$", B, dir(O--B));<br />
label("$C$", C, dir(O--C));<br />
label("$D$", D, dir(O--D));<br />
label("$F$", F, dir(O--F));<br />
label("$O$", O, NW);<br />
label("$E$", E, SE);</asy><br />
<br />
<math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math><br />
<br />
[[1995 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.<br />
<br />
<cmath>\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\<br />
& & & & 1 & & 1 & & & & \\<br />
& & & 2 & & 2 & & 2 & & & \\<br />
& & 3 & & 4 & & 4 & & 3 & & \\<br />
& 4 & & 7 & & 8 & & 7 & & 4 & \\<br />
5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular}</cmath><br />
<br />
Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } </math><br />
<br />
[[1995 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is<br />
<asy><br />
// note: diagram deliberately not to scale -- azjps<br />
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }<br />
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
real min = -0.6, step = 0.5;<br />
pair[] A, B; D(unitcircle);<br />
for(int i = 0; i < 3; ++i) {<br />
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);<br />
D(D(A[i])--D(B[i]));<br />
}<br />
MP("10",(A[0]+B[0])/2,N);<br />
MP("\sqrt{a}",(A[1]+B[1])/2,N);<br />
MP("14",(A[2]+B[2])/2,N);<br />
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);<br />
</asy><br />
<math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } </math><br />
<br />
[[1995 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?<br />
<br />
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 } </math><br />
<br />
[[1995 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is<br />
<asy><br />
size(150); defaultpen(linewidth(0.7)); pair slant = (2,1); <br />
for(int i = 0; i < 4; ++i) <br />
draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); <br />
for(int i = 1; i < 4; ++i)<br />
draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);<br />
</asy><br />
<math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } </math><br />
<br />
[[1995 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
{{AHSME box|year=1995|before=[[1994 AHSME Problems|1994 AHSME]]|after=[[1996 AHSME Problems|1996 AHSME]]}}<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems&diff=446471995 AHSME Problems2012-02-10T02:21:27Z<p>Freddylukai: /* Problem 6 */</p>
<hr />
<div>== Problem 1 ==<br />
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <br />
<br />
<math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qquad \mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by 3} } \qquad \mathrm{(E) \ \text{increase by 4} } </math><br />
<br />
[[1995 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math><br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \sqrt{7} } \qquad \mathrm{(C) \ 7 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 121 } </math><br />
<br />
[[1995 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How many cents are saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
If <math>M</math> is <math>30 \%</math> of <math>Q</math>, <math>Q</math> is <math>20 \%</math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math><br />
<br />
<math> \mathrm{(A) \ \frac {3}{250} } \qquad \mathrm{(B) \ \frac {3}{25} } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ \frac {6}{5} } \qquad \mathrm{(E) \ \frac {4}{3} } </math><br />
<br />
[[1995 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is <br />
<br />
<math> \mathrm{(A) \ \text{500 thousand} } \qquad \mathrm{(B) \ \text{5 million} } \qquad \mathrm{(C) \ \text{50 million} } \qquad \mathrm{(D) \ \text{500 million} } \qquad \mathrm{(E) \ \text{5 billion} } </math><br />
<br />
[[1995 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked x? <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);<br />
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);<br />
draw(shift(1,0)*p, dashed);<br />
label("$x$", (0.3,0.5), E);<br />
label("$A$", (1.3,0.5), E);<br />
label("$B$", (1.3,1.5), E);<br />
label("$C$", (2.3,1.5), E);<br />
label("$D$", (2.3,2.5), E);<br />
label("$E$", (3.3,2.5), E);</asy><br />
<br />
<math> \mathrm{(A) \ A } \qquad \mathrm{(B) \ B } \qquad \mathrm{(C) \ C } \qquad \mathrm{(D) \ D } \qquad \mathrm{(E) \ E } </math><br />
<br />
[[1995 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: <br />
<br />
<math> \mathrm{(A) \ 8 } \qquad \mathrm{(B) \ 25 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 75 } \qquad \mathrm{(E) \ 100 } </math><br />
<br />
[[1995 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
In <math>\triangle ABC</math>, <math>\angle C = 90^\circ, AC = 6</math> and <math>BC = 8</math>. Points <math>D</math> and <math>E</math> are on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, and <math>\angle BED = 90^\circ</math>. If <math>DE = 4</math>, then <math>BD =</math><br />
<asy> <br />
size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16)); <br />
</asy><br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ \frac {16}{3} } \qquad \mathrm{(C) \ \frac {20}{3} } \qquad \mathrm{(D) \ \frac {15}{2} } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is <br />
<asy><br />
size(100); defaultpen(linewidth(0.7)); draw(unitsquare^^(0,0)--(1,1)^^(0,1)--(1,0)^^(.5,0)--(.5,1)^^(0,.5)--(1,.5));<br />
</asy><br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 12 } \qquad \mathrm{(C) \ 14 } \qquad \mathrm{(D) \ 16 } \qquad \mathrm{(E) \ 18 } </math><br />
<br />
[[1995 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The area of the triangle bounded by the lines <math>y = x, y = - x</math> and <math>y = 6</math> is<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 12\sqrt{2} } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 24\sqrt{2} } \qquad \mathrm{(E) \ 36 } </math><br />
<br />
[[1995 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
How many base 10 four-digit numbers, <math>N = \underline{a} \underline{b} \underline{c} \underline{d}</math>, satisfy all three of the following conditions?<br />
<br />
(i) <math>4,000 \leq N < 6,000;</math> (ii) <math>N</math> is a multiple of 5; (iii) <math>3 \leq b < c \leq 6</math>.<br />
<br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 18 } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 36 } \qquad \mathrm{(E) \ 48 } </math><br />
<br />
[[1995 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
Let <math>f</math> be a linear function with the properties that <math>f(1) \leq f(2), f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true?<br />
<br />
<math> \mathrm{(A) \ f(0) < 0 } \qquad \mathrm{(B) \ f(0) = 0 } \qquad \mathrm{(C) \ f(1) < f(0) < f( - 1) } \qquad \mathrm{(D) \ f(0) = 5 } \qquad \mathrm{(E) \ f(0) > 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit <math>e</math>. Find the sum of <math>d</math> and <math>e</math>.<br />
<br />
<math>\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\<br />
+ & 8 & 2 & 9 & 4 & 3 & 0 \\<br />
\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}</math><br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ \text{more than 10} } </math><br />
<br />
[[1995 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
If <math>f(x) = ax^4 - bx^2 + x + 5</math> and <math>f( - 3) = 2</math>, then <math>f(3) =</math><br />
<br />
<math> \mathrm{(A) \ -5 } \qquad \mathrm{(B) \ -2 } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point <br />
<asy><br />
size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle);<br />
for(int i = 0; i < 5; ++i) { pair P = dir(90-i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }<br />
</asy><br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:<br />
<br />
i. The actual attendance in Atlanta is within <math>10 \%</math> of Anita's estimate.<br />
ii. Bob's estimate is within <math>10 \%</math> of the actual attendance in Boston.<br />
<br />
To the nearest 1,000, the largest possible difference between the numbers attending the two games is<br />
<br />
<math> \mathrm{(A) \ 10000 } \qquad \mathrm{(B) \ 11000 } \qquad \mathrm{(C) \ 20000 } \qquad \mathrm{(D) \ 21000 } \qquad \mathrm{(E) \ 22000 } </math><br />
<br />
[[1995 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is<br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(rotate(18)*polygon(5));<br />
real x=0.6180339887;<br />
draw(Circle((-x,0), 1));<br />
int i;<br />
for(i=0; i<5; i=i+1) {<br />
dot(origin+1*dir(36+72*i));<br />
}<br />
label("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0)));<br />
label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72)));<br />
label("$E$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144)));<br />
label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3)));<br />
label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));</asy><br />
<br />
<math> \mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 } </math><br />
<br />
[[1995 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
Two rays with common endpoint <math>O</math> forms a <math>30^\circ</math> angle. Point <math>A</math> lies on one ray, point <math>B</math> on the other ray, and <math>AB = 1</math>. The maximum possible length of <math>OB</math> is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \frac {1 + \sqrt {3}}{\sqrt 2} } \qquad \mathrm{(C) \ \sqrt{3} } \qquad \mathrm{(D) \ 2 } \qquad \mathrm{(E) \ \frac{4}{\sqrt{3}} } </math><br />
<br />
[[1995 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
Equilateral triangle <math>DEF</math> is inscribed in equilateral triangle <math>ABC</math> such that <math>\overline{DE} \perp \overline{BC}</math>. The reatio of the area of <math>\triangle DEF</math> to the area of <math>\triangle ABC</math> is<br />
<asy><br />
pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);<br />
pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3;<br />
D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));<br />
</asy><br />
<math> \mathrm{(A) \ \frac {1}{6} } \qquad \mathrm{(B) \ \frac {1}{4} } \qquad \mathrm{(C) \ \frac {1}{3} } \qquad \mathrm{(D) \ \frac {2}{5} } \qquad \mathrm{(E) \ \frac {1}{2} } </math><br />
<br />
[[1995 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
If <math>a,b</math> and <math>c</math> are three (not necessarily different) numbers chosen randomly and with replacement from the set <math>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is<br />
<br />
<math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } </math><br />
<br />
[[1995 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
Two nonadjacent vertices of a rectangle are <math>(4,3)</math> and <math>(-4,-3)</math>, and the coordinates of the other two vertices are integers. The number of such rectangles is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is<br />
<br />
<math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 } </math><br />
<br />
[[1995 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The sides of a triangle have lengths 11,15, and <math>k</math>, where <math>k</math> is an integer. For how many values of <math>k</math> is the triangle obtuse?<br />
<br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 } </math><br />
<br />
[[1995 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that<br />
<br />
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath><br />
<br />
What is <math>A + B + C</math>?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list? <br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } </math><br />
<br />
[[1995 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(Circle(origin, 5));<br />
pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D);<br />
draw(A--B^^C--D--F);<br />
dot(O^^A^^B^^C^^D^^E^^F);<br />
markscalefactor=0.05;<br />
draw(rightanglemark(B, O, D));<br />
label("$A$", A, dir(O--A));<br />
label("$B$", B, dir(O--B));<br />
label("$C$", C, dir(O--C));<br />
label("$D$", D, dir(O--D));<br />
label("$F$", F, dir(O--F));<br />
label("$O$", O, NW);<br />
label("$E$", E, SE);</asy><br />
<br />
<math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math><br />
<br />
[[1995 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.<br />
<br />
<cmath>\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\<br />
& & & & 1 & & 1 & & & & \\<br />
& & & 2 & & 2 & & 2 & & & \\<br />
& & 3 & & 4 & & 4 & & 3 & & \\<br />
& 4 & & 7 & & 8 & & 7 & & 4 & \\<br />
5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular}</cmath><br />
<br />
Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } </math><br />
<br />
[[1995 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is<br />
<asy><br />
// note: diagram deliberately not to scale -- azjps<br />
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }<br />
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
real min = -0.6, step = 0.5;<br />
pair[] A, B; D(unitcircle);<br />
for(int i = 0; i < 3; ++i) {<br />
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);<br />
D(D(A[i])--D(B[i]));<br />
}<br />
MP("10",(A[0]+B[0])/2,N);<br />
MP("\sqrt{a}",(A[1]+B[1])/2,N);<br />
MP("14",(A[2]+B[2])/2,N);<br />
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);<br />
</asy><br />
<math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } </math><br />
<br />
[[1995 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?<br />
<br />
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 } </math><br />
<br />
[[1995 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is<br />
<asy><br />
size(150); defaultpen(linewidth(0.7)); pair slant = (2,1); <br />
for(int i = 0; i < 4; ++i) <br />
draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); <br />
for(int i = 1; i < 4; ++i)<br />
draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);<br />
</asy><br />
<math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } </math><br />
<br />
[[1995 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
{{AHSME box|year=1995|before=[[1994 AHSME Problems|1994 AHSME]]|after=[[1996 AHSME Problems|1996 AHSME]]}}<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems&diff=446461995 AHSME Problems2012-02-10T02:21:17Z<p>Freddylukai: /* Problem 6 */</p>
<hr />
<div>== Problem 1 ==<br />
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <br />
<br />
<math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qquad \mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by 3} } \qquad \mathrm{(E) \ \text{increase by 4} } </math><br />
<br />
[[1995 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math><br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \sqrt{7} } \qquad \mathrm{(C) \ 7 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 121 } </math><br />
<br />
[[1995 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How many cents are saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
If <math>M</math> is <math>30 \%</math> of <math>Q</math>, <math>Q</math> is <math>20 \%</math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math><br />
<br />
<math> \mathrm{(A) \ \frac {3}{250} } \qquad \mathrm{(B) \ \frac {3}{25} } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ \frac {6}{5} } \qquad \mathrm{(E) \ \frac {4}{3} } </math><br />
<br />
[[1995 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is <br />
<br />
<math> \mathrm{(A) \ \text{500 thousand} } \qquad \mathrm{(B) \ \text{5 million} } \qquad \mathrm{(C) \ \text{50 million} } \qquad \mathrm{(D) \ \text{500 million} } \qquad \mathrm{(E) \ \text{5 billion} } </math><br />
<br />
[[1995 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked x <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);<br />
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);<br />
draw(shift(1,0)*p, dashed);<br />
label("$x$", (0.3,0.5), E);<br />
label("$A$", (1.3,0.5), E);<br />
label("$B$", (1.3,1.5), E);<br />
label("$C$", (2.3,1.5), E);<br />
label("$D$", (2.3,2.5), E);<br />
label("$E$", (3.3,2.5), E);</asy><br />
<br />
<math> \mathrm{(A) \ A } \qquad \mathrm{(B) \ B } \qquad \mathrm{(C) \ C } \qquad \mathrm{(D) \ D } \qquad \mathrm{(E) \ E } </math><br />
<br />
[[1995 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: <br />
<br />
<math> \mathrm{(A) \ 8 } \qquad \mathrm{(B) \ 25 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 75 } \qquad \mathrm{(E) \ 100 } </math><br />
<br />
[[1995 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
In <math>\triangle ABC</math>, <math>\angle C = 90^\circ, AC = 6</math> and <math>BC = 8</math>. Points <math>D</math> and <math>E</math> are on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, and <math>\angle BED = 90^\circ</math>. If <math>DE = 4</math>, then <math>BD =</math><br />
<asy> <br />
size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16)); <br />
</asy><br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ \frac {16}{3} } \qquad \mathrm{(C) \ \frac {20}{3} } \qquad \mathrm{(D) \ \frac {15}{2} } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is <br />
<asy><br />
size(100); defaultpen(linewidth(0.7)); draw(unitsquare^^(0,0)--(1,1)^^(0,1)--(1,0)^^(.5,0)--(.5,1)^^(0,.5)--(1,.5));<br />
</asy><br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 12 } \qquad \mathrm{(C) \ 14 } \qquad \mathrm{(D) \ 16 } \qquad \mathrm{(E) \ 18 } </math><br />
<br />
[[1995 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The area of the triangle bounded by the lines <math>y = x, y = - x</math> and <math>y = 6</math> is<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 12\sqrt{2} } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 24\sqrt{2} } \qquad \mathrm{(E) \ 36 } </math><br />
<br />
[[1995 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
How many base 10 four-digit numbers, <math>N = \underline{a} \underline{b} \underline{c} \underline{d}</math>, satisfy all three of the following conditions?<br />
<br />
(i) <math>4,000 \leq N < 6,000;</math> (ii) <math>N</math> is a multiple of 5; (iii) <math>3 \leq b < c \leq 6</math>.<br />
<br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 18 } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 36 } \qquad \mathrm{(E) \ 48 } </math><br />
<br />
[[1995 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
Let <math>f</math> be a linear function with the properties that <math>f(1) \leq f(2), f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true?<br />
<br />
<math> \mathrm{(A) \ f(0) < 0 } \qquad \mathrm{(B) \ f(0) = 0 } \qquad \mathrm{(C) \ f(1) < f(0) < f( - 1) } \qquad \mathrm{(D) \ f(0) = 5 } \qquad \mathrm{(E) \ f(0) > 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit <math>e</math>. Find the sum of <math>d</math> and <math>e</math>.<br />
<br />
<math>\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\<br />
+ & 8 & 2 & 9 & 4 & 3 & 0 \\<br />
\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}</math><br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ \text{more than 10} } </math><br />
<br />
[[1995 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
If <math>f(x) = ax^4 - bx^2 + x + 5</math> and <math>f( - 3) = 2</math>, then <math>f(3) =</math><br />
<br />
<math> \mathrm{(A) \ -5 } \qquad \mathrm{(B) \ -2 } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point <br />
<asy><br />
size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle);<br />
for(int i = 0; i < 5; ++i) { pair P = dir(90-i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }<br />
</asy><br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:<br />
<br />
i. The actual attendance in Atlanta is within <math>10 \%</math> of Anita's estimate.<br />
ii. Bob's estimate is within <math>10 \%</math> of the actual attendance in Boston.<br />
<br />
To the nearest 1,000, the largest possible difference between the numbers attending the two games is<br />
<br />
<math> \mathrm{(A) \ 10000 } \qquad \mathrm{(B) \ 11000 } \qquad \mathrm{(C) \ 20000 } \qquad \mathrm{(D) \ 21000 } \qquad \mathrm{(E) \ 22000 } </math><br />
<br />
[[1995 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is<br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(rotate(18)*polygon(5));<br />
real x=0.6180339887;<br />
draw(Circle((-x,0), 1));<br />
int i;<br />
for(i=0; i<5; i=i+1) {<br />
dot(origin+1*dir(36+72*i));<br />
}<br />
label("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0)));<br />
label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72)));<br />
label("$E$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144)));<br />
label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3)));<br />
label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));</asy><br />
<br />
<math> \mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 } </math><br />
<br />
[[1995 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
Two rays with common endpoint <math>O</math> forms a <math>30^\circ</math> angle. Point <math>A</math> lies on one ray, point <math>B</math> on the other ray, and <math>AB = 1</math>. The maximum possible length of <math>OB</math> is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \frac {1 + \sqrt {3}}{\sqrt 2} } \qquad \mathrm{(C) \ \sqrt{3} } \qquad \mathrm{(D) \ 2 } \qquad \mathrm{(E) \ \frac{4}{\sqrt{3}} } </math><br />
<br />
[[1995 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
Equilateral triangle <math>DEF</math> is inscribed in equilateral triangle <math>ABC</math> such that <math>\overline{DE} \perp \overline{BC}</math>. The reatio of the area of <math>\triangle DEF</math> to the area of <math>\triangle ABC</math> is<br />
<asy><br />
pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);<br />
pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3;<br />
D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));<br />
</asy><br />
<math> \mathrm{(A) \ \frac {1}{6} } \qquad \mathrm{(B) \ \frac {1}{4} } \qquad \mathrm{(C) \ \frac {1}{3} } \qquad \mathrm{(D) \ \frac {2}{5} } \qquad \mathrm{(E) \ \frac {1}{2} } </math><br />
<br />
[[1995 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
If <math>a,b</math> and <math>c</math> are three (not necessarily different) numbers chosen randomly and with replacement from the set <math>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is<br />
<br />
<math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } </math><br />
<br />
[[1995 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
Two nonadjacent vertices of a rectangle are <math>(4,3)</math> and <math>(-4,-3)</math>, and the coordinates of the other two vertices are integers. The number of such rectangles is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is<br />
<br />
<math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 } </math><br />
<br />
[[1995 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The sides of a triangle have lengths 11,15, and <math>k</math>, where <math>k</math> is an integer. For how many values of <math>k</math> is the triangle obtuse?<br />
<br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 } </math><br />
<br />
[[1995 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that<br />
<br />
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath><br />
<br />
What is <math>A + B + C</math>?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list? <br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } </math><br />
<br />
[[1995 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(Circle(origin, 5));<br />
pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D);<br />
draw(A--B^^C--D--F);<br />
dot(O^^A^^B^^C^^D^^E^^F);<br />
markscalefactor=0.05;<br />
draw(rightanglemark(B, O, D));<br />
label("$A$", A, dir(O--A));<br />
label("$B$", B, dir(O--B));<br />
label("$C$", C, dir(O--C));<br />
label("$D$", D, dir(O--D));<br />
label("$F$", F, dir(O--F));<br />
label("$O$", O, NW);<br />
label("$E$", E, SE);</asy><br />
<br />
<math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math><br />
<br />
[[1995 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.<br />
<br />
<cmath>\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\<br />
& & & & 1 & & 1 & & & & \\<br />
& & & 2 & & 2 & & 2 & & & \\<br />
& & 3 & & 4 & & 4 & & 3 & & \\<br />
& 4 & & 7 & & 8 & & 7 & & 4 & \\<br />
5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular}</cmath><br />
<br />
Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } </math><br />
<br />
[[1995 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is<br />
<asy><br />
// note: diagram deliberately not to scale -- azjps<br />
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }<br />
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
real min = -0.6, step = 0.5;<br />
pair[] A, B; D(unitcircle);<br />
for(int i = 0; i < 3; ++i) {<br />
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);<br />
D(D(A[i])--D(B[i]));<br />
}<br />
MP("10",(A[0]+B[0])/2,N);<br />
MP("\sqrt{a}",(A[1]+B[1])/2,N);<br />
MP("14",(A[2]+B[2])/2,N);<br />
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);<br />
</asy><br />
<math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } </math><br />
<br />
[[1995 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?<br />
<br />
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 } </math><br />
<br />
[[1995 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is<br />
<asy><br />
size(150); defaultpen(linewidth(0.7)); pair slant = (2,1); <br />
for(int i = 0; i < 4; ++i) <br />
draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); <br />
for(int i = 1; i < 4; ++i)<br />
draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);<br />
</asy><br />
<math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } </math><br />
<br />
[[1995 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
{{AHSME box|year=1995|before=[[1994 AHSME Problems|1994 AHSME]]|after=[[1996 AHSME Problems|1996 AHSME]]}}<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems/Problem_3&diff=446451995 AHSME Problems/Problem 32012-02-10T02:18:21Z<p>Freddylukai: /* Problem */ added solution</p>
<hr />
<div>== Problem ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How many cents are saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
<br />
== Solution ==<br />
IMPORTANT NOTICE: The original problem statement had "how much is saved". However, because this made little sense when the calculations were done, the problem statement was changed to "how many cents".<br />
<br />
We see that 3 payments of <math>\textdollar 29.98</math> will be a total cost of <math>3\cdot(30-.02)=90-.06</math><br />
<br />
Adding this to <math>\textdollar9.98</math> we have a total of <math>99.98-.06</math><br />
<br />
Clearly, this differs from <math>\textdollar 99.99</math> by <math>7</math> cents. Thus, the answer is <math>\fbox{\text{(B)}}</math><br />
<br />
== See also ==<br />
{{AHSME box|year=1995|num-b=2|num-a=4}}<br />
<br />
[[Category:Introductory Algebra Problems]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems/Problem_3&diff=446441995 AHSME Problems/Problem 32012-02-10T02:13:53Z<p>Freddylukai: /* Problem */</p>
<hr />
<div>== Problem ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How many cents are saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
== See also ==<br />
{{AHSME box|year=1995|num-b=2|num-a=4}}<br />
<br />
[[Category:Introductory Algebra Problems]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems&diff=446431995 AHSME Problems2012-02-10T02:13:35Z<p>Freddylukai: /* Problem 3 */ edit to what the problem probably originally stated</p>
<hr />
<div>== Problem 1 ==<br />
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <br />
<br />
<math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qquad \mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by 3} } \qquad \mathrm{(E) \ \text{increase by 4} } </math><br />
<br />
[[1995 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math><br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \sqrt{7} } \qquad \mathrm{(C) \ 7 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 121 } </math><br />
<br />
[[1995 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How many cents are saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
If <math>M</math> is <math>30 \%</math> of <math>Q</math>, <math>Q</math> is <math>20 \%</math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math><br />
<br />
<math> \mathrm{(A) \ \frac {3}{250} } \qquad \mathrm{(B) \ \frac {3}{25} } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ \frac {6}{5} } \qquad \mathrm{(E) \ \frac {4}{3} } </math><br />
<br />
[[1995 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is <br />
<br />
<math> \mathrm{(A) \ \text{500 thousand} } \qquad \mathrm{(B) \ \text{5 million} } \qquad \mathrm{(C) \ \text{50 million} } \qquad \mathrm{(D) \ \text{500 million} } \qquad \mathrm{(E) \ \text{5 billion} } </math><br />
<br />
[[1995 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked ? <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);<br />
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);<br />
draw(shift(1,0)*p, dashed);<br />
label("$x$", (0.3,0.5), E);<br />
label("$A$", (1.3,0.5), E);<br />
label("$B$", (1.3,1.5), E);<br />
label("$C$", (2.3,1.5), E);<br />
label("$D$", (2.3,2.5), E);<br />
label("$E$", (3.3,2.5), E);</asy><br />
<br />
<math> \mathrm{(A) \ A } \qquad \mathrm{(B) \ B } \qquad \mathrm{(C) \ C } \qquad \mathrm{(D) \ D } \qquad \mathrm{(E) \ E } </math><br />
<br />
[[1995 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: <br />
<br />
<math> \mathrm{(A) \ 8 } \qquad \mathrm{(B) \ 25 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 75 } \qquad \mathrm{(E) \ 100 } </math><br />
<br />
[[1995 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
In <math>\triangle ABC</math>, <math>\angle C = 90^\circ, AC = 6</math> and <math>BC = 8</math>. Points <math>D</math> and <math>E</math> are on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, and <math>\angle BED = 90^\circ</math>. If <math>DE = 4</math>, then <math>BD =</math><br />
<asy> <br />
size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16)); <br />
</asy><br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ \frac {16}{3} } \qquad \mathrm{(C) \ \frac {20}{3} } \qquad \mathrm{(D) \ \frac {15}{2} } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is <br />
<asy><br />
size(100); defaultpen(linewidth(0.7)); draw(unitsquare^^(0,0)--(1,1)^^(0,1)--(1,0)^^(.5,0)--(.5,1)^^(0,.5)--(1,.5));<br />
</asy><br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 12 } \qquad \mathrm{(C) \ 14 } \qquad \mathrm{(D) \ 16 } \qquad \mathrm{(E) \ 18 } </math><br />
<br />
[[1995 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
The area of the triangle bounded by the lines <math>y = x, y = - x</math> and <math>y = 6</math> is<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 12\sqrt{2} } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 24\sqrt{2} } \qquad \mathrm{(E) \ 36 } </math><br />
<br />
[[1995 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
How many base 10 four-digit numbers, <math>N = \underline{a} \underline{b} \underline{c} \underline{d}</math>, satisfy all three of the following conditions?<br />
<br />
(i) <math>4,000 \leq N < 6,000;</math> (ii) <math>N</math> is a multiple of 5; (iii) <math>3 \leq b < c \leq 6</math>.<br />
<br />
<math> \mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 18 } \qquad \mathrm{(C) \ 24 } \qquad \mathrm{(D) \ 36 } \qquad \mathrm{(E) \ 48 } </math><br />
<br />
[[1995 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
Let <math>f</math> be a linear function with the properties that <math>f(1) \leq f(2), f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true?<br />
<br />
<math> \mathrm{(A) \ f(0) < 0 } \qquad \mathrm{(B) \ f(0) = 0 } \qquad \mathrm{(C) \ f(1) < f(0) < f( - 1) } \qquad \mathrm{(D) \ f(0) = 5 } \qquad \mathrm{(E) \ f(0) > 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit <math>e</math>. Find the sum of <math>d</math> and <math>e</math>.<br />
<br />
<math>\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\<br />
+ & 8 & 2 & 9 & 4 & 3 & 0 \\<br />
\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}</math><br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ \text{more than 10} } </math><br />
<br />
[[1995 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
If <math>f(x) = ax^4 - bx^2 + x + 5</math> and <math>f( - 3) = 2</math>, then <math>f(3) =</math><br />
<br />
<math> \mathrm{(A) \ -5 } \qquad \mathrm{(B) \ -2 } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 8 } </math><br />
<br />
[[1995 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point <br />
<asy><br />
size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle);<br />
for(int i = 0; i < 5; ++i) { pair P = dir(90-i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }<br />
</asy><br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:<br />
<br />
i. The actual attendance in Atlanta is within <math>10 \%</math> of Anita's estimate.<br />
ii. Bob's estimate is within <math>10 \%</math> of the actual attendance in Boston.<br />
<br />
To the nearest 1,000, the largest possible difference between the numbers attending the two games is<br />
<br />
<math> \mathrm{(A) \ 10000 } \qquad \mathrm{(B) \ 11000 } \qquad \mathrm{(C) \ 20000 } \qquad \mathrm{(D) \ 21000 } \qquad \mathrm{(E) \ 22000 } </math><br />
<br />
[[1995 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is<br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(rotate(18)*polygon(5));<br />
real x=0.6180339887;<br />
draw(Circle((-x,0), 1));<br />
int i;<br />
for(i=0; i<5; i=i+1) {<br />
dot(origin+1*dir(36+72*i));<br />
}<br />
label("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0)));<br />
label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72)));<br />
label("$E$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144)));<br />
label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3)));<br />
label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));</asy><br />
<br />
<math> \mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 } </math><br />
<br />
[[1995 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
Two rays with common endpoint <math>O</math> forms a <math>30^\circ</math> angle. Point <math>A</math> lies on one ray, point <math>B</math> on the other ray, and <math>AB = 1</math>. The maximum possible length of <math>OB</math> is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ \frac {1 + \sqrt {3}}{\sqrt 2} } \qquad \mathrm{(C) \ \sqrt{3} } \qquad \mathrm{(D) \ 2 } \qquad \mathrm{(E) \ \frac{4}{\sqrt{3}} } </math><br />
<br />
[[1995 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
Equilateral triangle <math>DEF</math> is inscribed in equilateral triangle <math>ABC</math> such that <math>\overline{DE} \perp \overline{BC}</math>. The reatio of the area of <math>\triangle DEF</math> to the area of <math>\triangle ABC</math> is<br />
<asy><br />
pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);<br />
pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3;<br />
D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));<br />
</asy><br />
<math> \mathrm{(A) \ \frac {1}{6} } \qquad \mathrm{(B) \ \frac {1}{4} } \qquad \mathrm{(C) \ \frac {1}{3} } \qquad \mathrm{(D) \ \frac {2}{5} } \qquad \mathrm{(E) \ \frac {1}{2} } </math><br />
<br />
[[1995 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
If <math>a,b</math> and <math>c</math> are three (not necessarily different) numbers chosen randomly and with replacement from the set <math>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is<br />
<br />
<math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } </math><br />
<br />
[[1995 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
Two nonadjacent vertices of a rectangle are <math>(4,3)</math> and <math>(-4,-3)</math>, and the coordinates of the other two vertices are integers. The number of such rectangles is<br />
<br />
<math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math><br />
<br />
[[1995 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is<br />
<br />
<math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 } </math><br />
<br />
[[1995 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
The sides of a triangle have lengths 11,15, and <math>k</math>, where <math>k</math> is an integer. For how many values of <math>k</math> is the triangle obtuse?<br />
<br />
<math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 } </math><br />
<br />
[[1995 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that<br />
<br />
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath><br />
<br />
What is <math>A + B + C</math>?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
[[1995 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list? <br />
<br />
<math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } </math><br />
<br />
[[1995 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is <br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(Circle(origin, 5));<br />
pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D);<br />
draw(A--B^^C--D--F);<br />
dot(O^^A^^B^^C^^D^^E^^F);<br />
markscalefactor=0.05;<br />
draw(rightanglemark(B, O, D));<br />
label("$A$", A, dir(O--A));<br />
label("$B$", B, dir(O--B));<br />
label("$C$", C, dir(O--C));<br />
label("$D$", D, dir(O--D));<br />
label("$F$", F, dir(O--F));<br />
label("$O$", O, NW);<br />
label("$E$", E, SE);</asy><br />
<br />
<math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math><br />
<br />
[[1995 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.<br />
<br />
<cmath>\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\<br />
& & & & 1 & & 1 & & & & \\<br />
& & & 2 & & 2 & & 2 & & & \\<br />
& & 3 & & 4 & & 4 & & 3 & & \\<br />
& 4 & & 7 & & 8 & & 7 & & 4 & \\<br />
5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular}</cmath><br />
<br />
Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?<br />
<br />
<math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } </math><br />
<br />
[[1995 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is<br />
<asy><br />
// note: diagram deliberately not to scale -- azjps<br />
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }<br />
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);<br />
real min = -0.6, step = 0.5;<br />
pair[] A, B; D(unitcircle);<br />
for(int i = 0; i < 3; ++i) {<br />
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);<br />
D(D(A[i])--D(B[i]));<br />
}<br />
MP("10",(A[0]+B[0])/2,N);<br />
MP("\sqrt{a}",(A[1]+B[1])/2,N);<br />
MP("14",(A[2]+B[2])/2,N);<br />
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);<br />
</asy><br />
<math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } </math><br />
<br />
[[1995 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?<br />
<br />
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 } </math><br />
<br />
[[1995 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is<br />
<asy><br />
size(150); defaultpen(linewidth(0.7)); pair slant = (2,1); <br />
for(int i = 0; i < 4; ++i) <br />
draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); <br />
for(int i = 1; i < 4; ++i)<br />
draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);<br />
</asy><br />
<math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } </math><br />
<br />
[[1995 AHSME Problems/Problem 30|Solution]]<br />
<br />
== See also ==<br />
{{AHSME box|year=1995|before=[[1994 AHSME Problems|1994 AHSME]]|after=[[1996 AHSME Problems|1996 AHSME]]}}<br />
* [[AHSME]]<br />
* [[AHSME Problems and Solutions]]<br />
* [[Mathematics competition resources]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1995_AHSME_Problems/Problem_3&diff=446421995 AHSME Problems/Problem 32012-02-10T02:12:55Z<p>Freddylukai: /* Problem */</p>
<hr />
<div>== Problem ==<br />
The total in-store price for an appliance is <math>\textdollar 99.99</math>. A television commercial advertises the same product for three easy payments of <math>\textdollar 29.98</math> and a one-time shipping and handling charge of <math>\textdollar 9.98</math>. How much is saved by buying the appliance from the television advertiser?<br />
<br />
<math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math><br />
<br />
== See also ==<br />
{{AHSME box|year=1995|num-b=2|num-a=4}}<br />
<br />
[[Category:Introductory Algebra Problems]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1989_AHSME_Problems&diff=446411989 AHSME Problems2012-02-10T02:07:33Z<p>Freddylukai: /* Problem 20 */ added problem</p>
<hr />
<div>== Problem 1 ==<br />
<br />
<math> (-1)^{5^{2}}+1^{2^{5}}= </math><br />
<br />
<math> \textrm{(A)}\ -7\qquad\textrm{(B)}\ -2\qquad\textrm{(C)}\ 0\qquad\textrm{(D)}\ 1\qquad\textrm{(E)}\ 57 </math><br />
<br />
[[1989 AHSME Problems/Problem 1|Solution]]<br />
<br />
== Problem 2 ==<br />
<br />
<math> \sqrt{\frac{1}{9}+\frac{1}{16}}= </math><br />
<br />
<math> \textrm{(A)}\ \frac{1}5\qquad\textrm{(B)}\ \frac{1}4\qquad\textrm{(C)}\ \frac{2}7\qquad\textrm{(D)}\ \frac{5}{12}\qquad\textrm{(E)}\ \frac{7}{12} </math><br />
<br />
[[1989 AHSME Problems/Problem 2|Solution]]<br />
<br />
== Problem 3 ==<br />
<br />
A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is<br />
<br />
<asy><br />
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);<br />
draw((3,0)--(3,9), dashed);<br />
draw((6,0)--(6,9), dashed);</asy><br />
<br />
<math> \textrm{(A)}\ 24\qquad\textrm{(B)}\ 36\qquad\textrm{(C)}\ 64\qquad\textrm{(D)}\ 81\qquad\textrm{(E)}\ 96 </math><br />
<br />
[[1989 AHSME Problems/Problem 3|Solution]]<br />
<br />
== Problem 4 ==<br />
<br />
In the figure, <math>ABCD</math> is an isosceles trapezoid with side lengths <math>AD=BC=5</math>, <math>AB=4</math>, and <math>DC=10</math>. The point <math>C</math> is on <math>\overline{DF}</math> and <math>B</math> is the midpoint of hypotenuse <math>\overline{DE}</math> in right triangle <math>DEF</math>. Then <math>CF=</math><br />
<br />
<asy><br />
defaultpen(fontsize(10));<br />
pair D=origin, A=(3,4), B=(7,4), C=(10,0), E=(14,8), F=(14,0);<br />
draw(B--C--F--E--B--A--D--B^^C--D, linewidth(0.7));<br />
dot(A^^B^^C^^D^^E^^F);<br />
pair point=(5,3);<br />
label("$A$", A, N);<br />
label("$B$", B, N);<br />
label("$C$", C, S);<br />
label("$D$", D, S);<br />
label("$E$", E, dir(point--E));<br />
label("$F$", F, dir(point--F));<br />
markscalefactor=0.05;<br />
draw(rightanglemark(E,F,D), linewidth(0.7));</asy><br />
<br />
<math> \textrm{(A)}\ 3.25\qquad\textrm{(B)}\ 3.5\qquad\textrm{(C)}\ 3.75\qquad\textrm{(D)}\ 4.0\qquad\textrm{(E)}\ 4.25 </math><br />
<br />
[[1989 AHSME Problems/Problem 4|Solution]]<br />
<br />
== Problem 5 ==<br />
<br />
Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is<br />
<br />
<asy><br />
real xscl = 1.2;<br />
int[] x = {0,1,2,4,5},y={0,1,3,4,5};<br />
for(int a:x){<br />
for(int b:y) {<br />
dot((a*xscl,b));<br />
}<br />
}<br />
for(int a:x) {<br />
pair prev = (a,y[0]);<br />
for(int i = 1;i<y.length;++i) {<br />
pair p = (a,y[i]);<br />
pen pen = linewidth(.7);<br />
if(y[i]-prev.y!=1){ <br />
pen+=dotted;<br />
}<br />
draw((xscl*prev.x,prev.y)--(xscl*p.x,p.y),pen);<br />
prev = p;<br />
}<br />
}for(int a:y) {<br />
pair prev = (x[0],a);<br />
for(int i = 1;i<x.length;++i) {<br />
pair p = (x[i],a);<br />
pen pen = linewidth(.7);<br />
if(x[i]-prev.x!=1){ <br />
pen+=dotted;<br />
}<br />
draw((xscl*prev.x,prev.y)--(p.x*xscl,p.y),pen);<br />
prev = p;<br />
}<br />
}<br />
path lblx = (0,-.7)--(5*xscl,-.7);<br />
draw(lblx);<br />
label("$10$",lblx);<br />
path lbly = (5*xscl+.7,0)--(5*xscl+.7,5);<br />
draw(lbly);<br />
label("$20$",lbly);</asy><br />
<br />
<math> \textrm{(A)}\ 30\qquad\textrm{(B)}\ 200\qquad\textrm{(C)}\ 410\qquad\textrm{(D)}\ 420\qquad\textrm{(E)}\ 430 </math><br />
<br />
[[1989 AHSME Problems/Problem 5|Solution]]<br />
<br />
== Problem 6 ==<br />
<br />
If <math>a,\,b>0</math> and the triangle in the first quadrant bounded by the coordinate axes and the graph of <math>ax+by=6</math> has area <math>6</math>, then <math>ab=</math><br />
<br />
<math> \textrm{(A)}\ 3\qquad\textrm{(B)}\ 6\qquad\textrm{(C)}\ 12\qquad\textrm{(D)}\ 108\qquad\textrm{(E)}\ 432 </math><br />
<br />
[[1989 AHSME Problems/Problem 6|Solution]]<br />
<br />
== Problem 7 ==<br />
<br />
In <math> \triangle ABC</math>, <math>\angle A = 100^\circ</math>, <math>\angle B = 50^\circ</math>, <math>\angle C = 30^\circ</math>, <math>\overline{AH}</math> is an altitude, and <math>\overline{BM}</math> is a median. Then <math>\angle MHC=</math><br />
<br />
<asy><br />
draw((0,0)--(16,0)--(6,6)--cycle);<br />
draw((6,6)--(6,0)--(11,3)--(0,0));<br />
dot((6,6));<br />
dot((0,0));<br />
dot((11,3));<br />
dot((6,0));<br />
dot((16,0));<br />
label("A", (6,6), N);<br />
label("B", (0,0), W);<br />
label("C", (16,0), E);<br />
label("H", (6,0), S);<br />
label("M", (11,3), NE);</asy><br />
<br />
<math> \textrm{(A)}\ 15^\circ\qquad\textrm{(B)}\ 22.5^\circ\qquad\textrm{(C)}\ 30^\circ\qquad\textrm{(D)}\ 40^\circ\qquad\textrm{(E)}\ 45^\circ </math><br />
<br />
[[1989 AHSME Problems/Problem 7|Solution]]<br />
<br />
== Problem 8 ==<br />
<br />
For how many integers <math>n</math> between <math>1</math> and <math>100</math> does <math> x^{2}+x-n </math> factor into the product of two linear factors with integer coefficients?<br />
<br />
<math> \textrm{(A)}\ 0\qquad\textrm{(B)}\ 1\qquad\textrm{(C)}\ 2\qquad\textrm{(D)}\ 9\qquad\textrm{(E)}\ 10 </math><br />
<br />
[[1989 AHSME Problems/Problem 8|Solution]]<br />
<br />
== Problem 9 ==<br />
<br />
Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible? <br />
<br />
<math> \textrm{(A)}\ 276\qquad\textrm{(B)}\ 300\qquad\textrm{(C)}\ 552\qquad\textrm{(D)}\ 600\qquad\textrm{(E)}\ 15600 </math><br />
<br />
[[1989 AHSME Problems/Problem 9|Solution]]<br />
<br />
== Problem 10 ==<br />
<br />
Consider the sequence defined recursively by <math> u_{1}= a </math> (any positive integer), and <math> u_{n+1}=\frac{-1}{u_{n}+1}</math>, <math>n = 1,2,3,\cdots </math>. For which of the following values of <math>n</math> must <math>u_{n}=a</math>?<br />
<br />
<math> \textrm{(A)}\ 14\qquad\textrm{(B)}\ 15\qquad\textrm{(C)}\ 16\qquad\textrm{(D)}\ 17\qquad\textrm{(E)}\ 18 </math><br />
<br />
[[1989 AHSME Problems/Problem 10|Solution]]<br />
<br />
== Problem 11 ==<br />
<br />
Let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> be positive integers with <math> a < 2b</math>, <math>b < 3c </math>, and <math>c<4d</math>. If <math>d<100</math>, the largest possible value for <math>a</math> is<br />
<br />
<math> \textrm{(A)}\ 2367\qquad\textrm{(B)}\ 2375\qquad\textrm{(C)}\ 2391\qquad\textrm{(D)}\ 2399\qquad\textrm{(E)}\ 2400 </math><br />
<br />
[[1989 AHSME Problems/Problem 11|Solution]]<br />
<br />
== Problem 12 ==<br />
<br />
The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a 100-mile section of highway?<br />
<br />
<math> \textrm{(A)}\ 100\qquad\textrm{(B)}\ 120\qquad\textrm{(C)}\ 200\qquad\textrm{(D)}\ 240\qquad\textrm{(E)}\ 400 </math><br />
<br />
[[1989 AHSME Problems/Problem 12|Solution]]<br />
<br />
== Problem 13 ==<br />
<br />
Two strips of width 1 overlap at an angle of <math>\alpha</math> as shown. The area of the overlap (shown shaded) is<br />
<br />
<asy><br />
pair a = (0,0),b= (6,0),c=(0,1),d=(6,1);<br />
transform t = rotate(-45,(3,.5));<br />
pair e = t*a,f=t*b,g=t*c,h=t*d;<br />
pair i = intersectionpoint(a--b,e--f),j=intersectionpoint(a--b,g--h),k=intersectionpoint(c--d,e--f),l=intersectionpoint(c--d,g--h);<br />
draw(a--b^^c--d^^e--f^^g--h);<br />
filldraw(i--j--l--k--cycle,blue);<br />
label("$\alpha$",i+(-.5,.2));<br />
//commented out labeling because it doesn't look right.<br />
//path lbl1 = (a+(.5,.2))--(c+(.5,-.2));<br />
//draw(lbl1);<br />
//label("$1$",lbl1);</asy><br />
<br />
<math> \textrm{(A)}\ \sin\alpha\qquad\textrm{(B)}\ \frac{1}{\sin\alpha}\qquad\textrm{(C)}\ \frac{1}{1-\cos\alpha}\qquad\textrm{(D)}\ \frac{1}{\sin^{2}\alpha}\qquad\textrm{(E)}\ \frac{1}{(1-\cos\alpha)^{2}} </math><br />
<br />
[[1989 AHSME Problems/Problem 13|Solution]]<br />
<br />
== Problem 14 ==<br />
<br />
<math> \cot 10+\tan 5 = </math><br />
<br />
<math> \textrm{(A)}\ \csc 5\qquad\textrm{(B)}\ \csc 10\qquad\textrm{(C)}\ \sec 5\qquad\textrm{(D)}\ \sec 10\qquad\textrm{(E)}\ \sin 15 </math><br />
<br />
[[1989 AHSME Problems/Problem 14|Solution]]<br />
<br />
== Problem 15 ==<br />
<br />
In <math>\triangle ABC</math>, <math>AB=5</math>, <math>BC=7</math>, <math>AC=9</math>, and <math>D</math> is on <math>\overline{AC}</math> with <math>BD=5</math>. Find the ratio of <math>AD:DC</math>. <br />
<br />
<asy><br />
draw((3,4)--(0,0)--(9,0)--(3,4)--(6,0));<br />
dot((0,0));<br />
dot((9,0));<br />
dot((3,4));<br />
dot((6,0));<br />
label("A", (0,0), W);<br />
label("B", (3,4), N);<br />
label("C", (9,0), E);<br />
label("D", (6,0), S);</asy><br />
<br />
<math> \textrm{(A)}\ 4:3\qquad\textrm{(B)}\ 7:5\qquad\textrm{(C)}\ 11:6\qquad\textrm{(D)}\ 13:5\qquad\textrm{(E)}\ 19:8 </math><br />
<br />
[[1989 AHSME Problems/Problem 15|Solution]]<br />
<br />
== Problem 16 ==<br />
<br />
A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.)<br />
<br />
<math> \textrm{(A)}\ 2\qquad\textrm{(B)}\ 4\qquad\textrm{(C)}\ 6\qquad\textrm{(D)}\ 16\qquad\textrm{(E)}\ 46 </math><br />
<br />
[[1989 AHSME Problems/Problem 16|Solution]]<br />
<br />
== Problem 17 ==<br />
<br />
The perimeter of an equilateral triangle exceeds the perimeter of a square by <math>1989</math> cm. The length of each side of the triangle exceeds the length of each side of the square by <math>d</math> cm. The square has perimeter greater than 0. How many positive integers are NOT a possible value for <math>d</math>?<br />
<br />
<math> \textrm{(A)}\ 0\qquad\textrm{(B)}\ 9\qquad\textrm{(C)}\ 221\qquad\textrm{(D)}\ 663\qquad\textrm{(E)}\ \text{infinitely many} </math><br />
<br />
[[1989 AHSME Problems/Problem 17|Solution]]<br />
<br />
== Problem 18 ==<br />
<br />
The set of all numbers x for which <math> x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}} </math> is a rational number is the set of all:<br />
<br />
<math> \textrm{(A)}\ \text{ integers }x\qquad\textrm{(B)}\ \text{ rational }x\qquad\textrm{(C)}\ \text{ real }x\qquad\textrm{(D)}\ x\text{ for which }\sqrt{x^{2}+1}\text{ is rational}\qquad\textrm{(E)}\ x\text{ for which }x+\sqrt{x^{2}+1}\text{ is rational } </math><br />
<br />
[[1989 AHSME Problems/Problem 18|Solution]]<br />
<br />
== Problem 19 ==<br />
<br />
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths <math>3</math>, <math>4</math>, and <math>5</math>. What is the area of the triangle?<br />
<br />
<math> \textrm{(A)}\ 6\qquad\textrm{(B)}\ \frac{18}{\pi^{2}}\qquad\textrm{(C)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}-1\right)\qquad\textrm{(D)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}+1\right)\qquad\textrm{(E)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}+3\right) </math><br />
<br />
[[1989 AHSME Problems/Problem 19|Solution]]<br />
<br />
== Problem 20 ==<br />
<br />
Let <math>x</math> be a real number selected uniformly at random between 100 and 200. If <math>\lfloor {\sqrt{x}} \rfloor = 12</math>, find the probability that <math>\lfloor {\sqrt{100x}} \rfloor = 120</math>. (<math>\lfloor {v} \rfloor</math> means the greatest integer less than or equal to <math>v</math>.)<br />
<br />
<math>\text{(A)} \ \frac{2}{25} \qquad \text{(B)} \ \frac{241}{2500} \qquad \text{(C)} \ \frac{1}{10} \qquad \text{(D)} \ \frac{96}{625} \qquad \text{(E)} \ 1</math><br />
<br />
<br />
[[1989 AHSME Problems/Problem 20|Solution]]<br />
<br />
== Problem 21 ==<br />
<br />
[[1989 AHSME Problems/Problem 21|Solution]]<br />
<br />
== Problem 22 ==<br />
<br />
[[1989 AHSME Problems/Problem 22|Solution]]<br />
<br />
== Problem 23 ==<br />
<br />
[[1989 AHSME Problems/Problem 23|Solution]]<br />
<br />
== Problem 24 ==<br />
<br />
[[1989 AHSME Problems/Problem 24|Solution]]<br />
<br />
== Problem 25 ==<br />
<br />
[[1989 AHSME Problems/Problem 25|Solution]]<br />
<br />
== Problem 26 ==<br />
<br />
[[1989 AHSME Problems/Problem 26|Solution]]<br />
<br />
== Problem 27 ==<br />
<br />
[[1989 AHSME Problems/Problem 27|Solution]]<br />
<br />
== Problem 28 ==<br />
<br />
[[1989 AHSME Problems/Problem 28|Solution]]<br />
<br />
== Problem 29 ==<br />
<br />
[[1989 AHSME Problems/Problem 29|Solution]]<br />
<br />
== Problem 30 ==<br />
<br />
[[1989 AHSME Problems/Problem 30|Solution]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems/Problem_16&diff=446351985 AHSME Problems/Problem 162012-02-10T00:34:50Z<p>Freddylukai: /* Solution */ added alternate solution</p>
<hr />
<div>==Problem==<br />
If <math> A=20^\circ </math> and <math> B=25^\circ </math>, then the value of <math> (1+\tan A)(1+\tan B) </math> is<br />
<br />
<math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2(\tan A+\tan B) \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
==Solution==<br />
First, let's leave everything in variables and see if we can simplify <math> (1+\tan A)(1+\tan B) </math>.<br />
<br />
<br />
We can write everything in terms of sine and cosine to get <math> \left(\frac{\cos A}{\cos A}+\frac{\sin A}{\cos A}\right)\left(\frac{\cos B}{\cos B}+\frac{\sin B}{\cos B}\right)=\frac{(\sin A+\cos A)(\sin B+\cos B)}{\cos A\cos B} </math>.<br />
<br />
<br />
<br />
We can multiply out the numerator to get <math> \frac{\sin A\sin B+\cos A\cos B+\sin A\cos B+\sin B\cos A}{\cos A\cos B} </math>.<br />
<br />
<br />
It may seem at first that we've made everything more complicated, however, we can recognize the numerator from the angle sum formulas:<br />
<br />
<br />
<math> \cos(A-B)=\sin A\sin B+\cos A\cos B </math><br />
<br />
<math> \sin(A+B)=\sin A\cos B+\sin B\cos A </math><br />
<br />
<br />
Therefore, our fraction is equal to <math> \frac{\cos(A-B)+\sin(A+B)}{\cos A\cos B} </math>.<br />
<br />
<br />
We can also use the product-to-sum formula<br />
<br />
<math> \cos A\cos B=\frac{1}{2}(\cos(A-B)+\cos(A+B)) </math> to simplify the denominator:<br />
<br />
<br />
<math> \frac{\cos(A-B)+\sin(A+B)}{\frac{1}{2}(\cos(A-B)+\cos(A+B))} </math>.<br />
<br />
<br />
But now we seem stuck. However, we can note that since <math> A+B=45^\circ </math>, we have <math> \sin(A+B)=\cos(A+B) </math>, so we get<br />
<br />
<br />
<math> \frac{\cos(A-B)+\sin(A+B)}{\frac{1}{2}(\cos(A-B)+\sin(A+B))} </math><br />
<br />
<br />
<math> \frac{1}{\frac{1}{2}} </math><br />
<br />
<math> 2, \boxed{\text{B}} </math><br />
<br />
Note that we only used the fact that <math> \sin(A+B)=\cos(A+B) </math>, so we have in fact not just shown that <math> (1+\tan A)(1+\tan B)=2 </math> for <math> A=20^\circ </math> and <math> B=25^\circ </math>, but for all <math> A, B </math> such that <math> A+B=45^\circ+n180^\circ </math>, for integer <math> n </math>.<br />
<br />
<br />
==Alternate Solution ==<br />
<br />
We can see that <math>25^o+20^o=45^o</math>. We also know that <math>\tan 45=1</math>. First, let us expand <math>(1+\tan A)(1+\tan B)</math>.<br />
<br />
We get <math>1+\tan A+\tan B+\tan A\tan B</math>. <br />
<br />
Now, let us look at <math>\tan45=\tan(20+25)</math>.<br />
<br />
By the <math>\tan</math> sum formula, we know that <math>\tan45=\dfrac{\text{tan A}+\text{tan B}}{1- \text{tan A} \text{tan B}}</math><br />
<br />
Then, since <math>\tan 45=1</math>, we can see that <math>\tan A+\tan B=1-\tan A\tan B</math><br />
<br />
Then <math>1=\tan A+\tan B+\tan A\tan B</math><br />
<br />
Thus, the sum become <math>1+1=2</math> and the answer is <math>\fbox{\text{(B)}}</math><br />
<br />
==See Also==<br />
{{AHSME box|year=1985|num-b=15|num-a=17}}</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems/Problem_11&diff=446321985 AHSME Problems/Problem 112012-02-10T00:18:10Z<p>Freddylukai: /* Solution */ changed answer</p>
<hr />
<div>==Problem==<br />
How many '''distinguishable''' rearrangements of the letters in CONTEST have both the vowels first? (For instance, OETCNST is one such arrangement but OTETSNC is not.)<br />
<br />
<math> \mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 720 \qquad \mathrm{(E) \ }2520 </math><br />
<br />
==Solution==<br />
We can separate each rearrangement into two parts: the vowels and the consonants. There are <math> 2 </math> possibilities for the first value and <math> 1 </math> for the remaining one, for a total of <math> 2\cdot1=2 </math> possible orderings of the vowels. There are <math> 5 </math> possibilities for the first consonant, <math> 4 </math> for the second, <math> 3 </math> for the third, <math> 2 </math> for the second, and <math> 1 </math> for the first, for a total of <math> 5\cdot4\cdot3\cdot2\cdot1=120 </math> possible orderings of the consonants. However, since both T's are indistinguishable, we must divide this total by <math>2!=2</math>. Thus, the actual number of total orderings of consonants is <math>120/2=60</math> In total, there are <math> 2\cdot60=120 </math> possible rearrangements, <math>\fbox{\text{(B)}}</math>.<br />
<br />
==See Also==<br />
{{AHSME box|year=1985|num-b=10|num-a=12}}</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems&diff=446301985 AHSME Problems2012-02-10T00:07:34Z<p>Freddylukai: /* Problem 12 */</p>
<hr />
<div>==Problem 1==<br />
If <math> 2x+1=8 </math>, then <math> 4x+1= </math><br />
<br />
<math> \mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{(D) \ } 18 \qquad \mathrm{(E) \ }19 </math><br />
<br />
[[1985 AHSME Problems/Problem 1|Solution]]<br />
==Problem 2==<br />
In an arcade game, the "monster" is the shaded sector of a [[circle]] of [[radius]] <math> 1 </math> cm, as shown in the figure. The missing piece (the mouth) has central [[angle]] <math> 60^\circ </math>. What is the [[perimeter]] of the monster in cm?<br />
<br />
<asy><br />
size(100);<br />
defaultpen(linewidth(0.7));<br />
filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black);<br />
label("1", (sqrt(3)/4, 1/4), NW);<br />
label("$60^\circ$", (1,0));</asy><br />
<br />
<math> \mathrm{(A)\ } \pi+2 \qquad \mathrm{(B) \ }2\pi \qquad \mathrm{(C) \ } \frac{5}{3}\pi \qquad \mathrm{(D) \ } \frac{5}{6}\pi+2 \qquad \mathrm{(E) \ }\frac{5}{3}\pi+2 </math><br />
<br />
[[1985 AHSME Problems/Problem 2|Solution]]<br />
==Problem 3==<br />
In right <math> \triangle ABC </math> with legs <math> 5 </math> and <math> 12 </math>, arcs of circles are drawn, one with center <math> A </math> and radius <math> 12 </math>, the other with center <math> B </math> and radius <math> 5 </math>. They intersect the [[hypotenuse]] at <math> M </math> and <math> N </math>. Then, <math> MN </math> has length: <br />
<br />
<asy><br />
defaultpen(linewidth(0.7)+fontsize(10));<br />
pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A);<br />
real r=degrees(B);<br />
draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270));<br />
pair point=incenter(A,B,C);<br />
label("$A$", A, dir(point--A));<br />
label("$B$", B, dir(point--B));<br />
label("$C$", C, dir(point--C));<br />
label("$M$", M, dir(point--M));<br />
label("$N$", N, dir(point--N));<br />
label("$12$", (6,0), S);<br />
label("$5$", (12,3.5), E);</asy><br />
<br />
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }\frac{13}{5} \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 4 \qquad \mathrm{(E) \ }\frac{24}{5} </math><br />
<br />
[[1985 AHSME Problems/Problem 3|Solution]]<br />
==Problem 4==<br />
A large bag of coins contains pennies, dimes, and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is<br />
<br />
<math> \mathrm{(A)\ } </math>&#036;<math>306 \qquad \mathrm{(B) \ } </math>&#036;<math>333 \qquad \mathrm{(C)\ } </math>&#036;<math>342 \qquad \mathrm{(D) \ } </math>&#036;<math>348 \qquad \mathrm{(E) \ } </math>&#036;<math>360 </math><br />
<br />
[[1985 AHSME Problems/Problem 4|Solution]]<br />
==Problem 5==<br />
Which terms must be removed from the sum<br />
<br />
<math> \frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12} </math><br />
<br />
if the sum of the remaining terms is equal to <math> 1 </math>?<br />
<br />
<math> \mathrm{(A)\ } \frac{1}{4}\text{ and }\frac{1}{8} \qquad \mathrm{(B) \ }\frac{1}{4}\text{ and }\frac{1}{12} \qquad \mathrm{(C) \ } \frac{1}{8}\text{ and }\frac{1}{12} \qquad \mathrm{(D) \ } \frac{1}{6}\text{ and }\frac{1}{10} \qquad \mathrm{(E) \ }\frac{1}{8}\text{ and }\frac{1}{10} </math><br />
<br />
[[1985 AHSME Problems/Problem 5|Solution]]<br />
==Problem 6==<br />
One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is <math> \frac{2}{3} </math> of the [[probability]] that a girl is chosen. The [[ratio]] of the number of boys to the total number of boys and girls is<br />
<br />
<math> \mathrm{(A)\ } \frac{1}{3} \qquad \mathrm{(B) \ }\frac{2}{5} \qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ } \frac{3}{5} \qquad \mathrm{(E) \ }\frac{2}{3} </math><br />
<br />
[[1985 AHSME Problems/Problem 6|Solution]]<br />
==Problem 7==<br />
In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from left to right. Thus, <math> a\times b-c </math> in such languages means the same as <math> a(b-c) </math> in ordinary algebraic notation. If <math> a\div b-c+d </math> is evaluated in such a language, the result in ordinary algebraic notation would be<br />
<br />
<math> \mathrm{(A)\ } \frac{a}{b}-c+d \qquad \mathrm{(B) \ }\frac{a}{b}-c-d \qquad \mathrm{(C) \ } \frac{d+c-b}{a} \qquad \mathrm{(D) \ } \frac{a}{b-c+d} \qquad \mathrm{(E) \ }\frac{a}{b-c-d} </math><br />
<br />
[[1985 AHSME Problems/Problem 7|Solution]]<br />
==Problem 8==<br />
Let <math> a, a', b, </math> and <math> b' </math> be real numbers with <math> a </math> and <math> a' </math> nonzero. The solution to <math> ax+b=0 </math> is less than the solution to <math> a'x+b'=0 </math> if and only if <br />
<br />
<math> \mathrm{(A)\ } a'b<ab' \qquad \mathrm{(B) \ }ab'<a'b \qquad \mathrm{(C) \ } ab<a'b' \qquad \mathrm{(D) \ } \frac{b}{a}<\frac{b'}{a'} \qquad </math><br />
<br />
<math> \mathrm{(E) \ }\frac{b'}{a'}<\frac{b}{a} </math><br />
<br />
[[1985 AHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
The odd positive integers <math> 1, 3, 5, 7, \cdots </math>, are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which <math> 1985 </math> appears in is the<br />
<br />
<asy><br />
int i,j;<br />
for(i=0; i<4; i=i+1) {<br />
label(string(16*i+1), (2*1,-2*i));<br />
label(string(16*i+3), (2*2,-2*i));<br />
label(string(16*i+5), (2*3,-2*i));<br />
label(string(16*i+7), (2*4,-2*i));<br />
}<br />
for(i=0; i<3; i=i+1) {<br />
for(j=0; j<4; j=j+1) {<br />
label(string(16*i+15-2*j), (2*j,-2*i-1));<br />
}}<br />
dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10));<br />
for(i=-10; i<-6; i=i+1) {<br />
for(j=1; j<4; j=j+1) {<br />
dot((2*j,i));<br />
}}</asy><br />
<br />
<math> \mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \ } \text{third} \qquad \mathrm{(D) \ } \text{fourth} \qquad \mathrm{(E) \ }\text{fifth} </math><br />
<br />
[[1985 AHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
An arbitrary [[circle]] can intersect the [[graph]] <math> y=\sin x </math> in<br />
<br />
<math> \mathrm{(A) } \text{at most }2\text{ points} \qquad \mathrm{(B) }\text{at most }4\text{ points} \qquad \mathrm{(C) } \text{at most }6\text{ points} \qquad \mathrm{(D) } \text{at most }8\text{ points}\qquad \mathrm{(E) }\text{more than }16\text{ points} </math><br />
<br />
[[1985 AHSME Problems/Problem 10|Solution]]<br />
==Problem 11==<br />
How many '''distinguishable''' rearrangements of the letters in CONTEST have both the vowels first? (For instance, OETCNST is one such arrangement but OTETSNC is not.)<br />
<br />
<math> \mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 720 \qquad \mathrm{(E) \ }2520 </math><br />
<br />
[[1985 AHSME Problems/Problem 11|Solution]]<br />
==Problem 12==<br />
Let's write <math> p, q, </math> and <math> r </math> as three distinct [[prime number]]s, where <math> 1 </math> is not a prime. Which of the following is the smallest positive [[perfect cube]] having <math> n=pq^2r^4 </math> as a [[divisor]]?<br />
<br />
<math> \mathrm{(A)\ } p^8q^8r^8 \qquad \mathrm{(B) \ }(pq^2r^2)^3 \qquad \mathrm{(C) \ } (p^2q^2r^2)^3 \qquad \mathrm{(D) \ } (pqr^2)^3 \qquad \mathrm{(E) \ }4p^3q^3r^3 </math><br />
<br />
[[1985 AHSME Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
Pegs are put in a board <math> 1 </math> unit apart both horizontally and vertically. A rubber band is stretched over <math> 4 </math> pegs as shown in the figure, forming a [[quadrilateral]]. Its [[area]] in square units is<br />
<br />
<asy><br />
int i,j;<br />
for(i=0; i<5; i=i+1) {<br />
for(j=0; j<4; j=j+1) {<br />
dot((i,j));<br />
}}<br />
draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7));</asy><br />
<br />
<math> \mathrm{(A)\ } 4 \qquad \mathrm{(B) \ }4.5 \qquad \mathrm{(C) \ } 5 \qquad \mathrm{(D) \ } 5.5 \qquad \mathrm{(E) \ }6 </math><br />
<br />
[[1985 AHSME Problems/Problem 13|Solution]]<br />
==Problem 14==<br />
Exactly three of the interior angles of a convex [[polygon]] are obtuse. What is the maximum number of sides of such a polygon?<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 />
[[1985 AHSME Problems/Problem 14|Solution]]<br />
==Problem 15==<br />
If <math> a </math> and <math> b </math> are positive numbers such that <math> a^b=b^a </math> and <math> b=9a </math>, then the value of <math> a </math> is:<br />
<br />
<math> \mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \ } \sqrt[9]{9} \qquad \mathrm{(D) \ } \sqrt[3]{9} \qquad \mathrm{(E) \ }\sqrt[4]{3} </math><br />
<br />
[[1985 AHSME Problems/Problem 15|Solution]]<br />
==Problem 16==<br />
If <math> A=20^\circ </math> and <math> B=25^\circ </math>, then the value of <math> (1+\tan A)(1+\tan B) </math> is<br />
<br />
<math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2(\tan A+\tan B) \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 16|Solution]]<br />
==Problem 17==<br />
[[Diagonal]] <math> DB </math> of [[rectangle]] <math> ABCD </math> is divided into <math> 3 </math> segments of length <math> 1 </math> by [[parallel]] lines <math> L </math> and <math> L' </math> that pass through <math> A </math> and <math> C </math> and are [[perpendicular]] to <math> DB </math>. The area of <math> ABCD </math>, rounded to the nearest tenth, is <br />
<br />
<asy><br />
defaultpen(linewidth(0.7)+fontsize(10));<br />
real x=sqrt(6), y=sqrt(3), a=0.4;<br />
pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D);<br />
real r=degrees(B);<br />
pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90);<br />
markscalefactor=0.02;<br />
draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B)^^rightanglemark(N1,E,B));<br />
pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135);<br />
label("A", A, NE);<br />
label("B", B, NE);<br />
label("C", C, dir(0));<br />
label("D", D, dir(180));<br />
label("$L$", (x/2,0), SW);<br />
label("$L^\prime$", C, SW);<br />
label("1", D--F, NW);<br />
label("1", F--E, SE);<br />
label("1", E--B, SE);<br />
clip(W--X--Y--Z--cycle);</asy><br />
<br />
<math> \mathrm{(A)\ } 4.1 \qquad \mathrm{(B) \ }4.2 \qquad \mathrm{(C) \ } 4.3 \qquad \mathrm{(D) \ } 4.4 \qquad \mathrm{(E) \ }4.5 </math><br />
<br />
[[1985 AHSME Problems/Problem 17|Solution]]<br />
==Problem 18==<br />
Six bags of marbles contain <math> 18, 19, 21, 23, 25, </math> and <math> 34 </math> marbles, respectively. One bag contains chipped marbles only. The other <math> 5 </math> bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?<br />
<br />
<math> \mathrm{(A)\ } 18 \qquad \mathrm{(B) \ }19 \qquad \mathrm{(C) \ } 21 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ }25 </math><br />
<br />
[[1985 AHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Consider the graphs <math> y=Ax^2 </math> and <math> y^2+3=x^2+4y </math>, where <math> A </math> is a positive constant and <math> x </math> and <math> y </math> are real variables. In how many points do the two graphs intersect?<br />
<br />
<math> \mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad </math> <br />
<br />
<math> \mathrm{(C) \ }\text{at least }1,\text{ but the number varies for different positive values of }A \qquad </math> <br />
<br />
<math> \mathrm{(D) \ }0\text{ for at least one positive value of }A \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 19|Solution]]<br />
==Problem 20==<br />
A wooden [[cube]] with edge length <math> n </math> units (where <math> n </math> is an integer <math> >2 </math>) is painted black all over. By slices parallel to its faces, the cube is cut into <math> n^3 </math> smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is <math> n </math>?<br />
<br />
<math> \mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 8 \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 20|Solution]]<br />
==Problem 21==<br />
How many integers <math> x </math> satisfy the equation <math> (x^2-x-1)^{x+2}=1 </math><br />
<br />
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 21|Solution]]<br />
==Problem 22==<br />
In a circle with center <math> O </math>, <math> AD </math> is a [[diameter]], <math> ABC </math> is a [[chord]], <math> BO=5 </math>, and <math> \angle ABO=\stackrel{\frown}{CD}=60^\circ </math>. Then the length of <math> BC </math> is:<br />
<br />
<asy><br />
defaultpen(linewidth(0.7)+fontsize(10));<br />
pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C);<br />
draw(C--A--D^^B--O^^Circle(O,1));<br />
pair point=O;<br />
label("$A$", A, dir(point--A));<br />
label("$B$", B, dir(point--B));<br />
label("$C$", C, dir(point--C));<br />
label("$D$", D, dir(point--D));<br />
label("$O$", O, dir(305));<br />
label("$5$", B--O, dir(O--B)*dir(90));<br />
label("$60^\circ$", dir(185), dir(185));<br />
label("$60^\circ$", B+0.05*dir(-25), dir(-25));</asy><br />
<br />
<math> \mathrm{(A)\ } 3 \qquad \mathrm{(B) \ }3+\sqrt{3} \qquad \mathrm{(C) \ } 5-\frac{\sqrt{3}}{2} \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of the above} </math><br />
<br />
[[1985 AHSME Problems/Problem 22|Solution]]<br />
==Problem 23==<br />
If <math> x=\frac{-1+i\sqrt{3}}{2} </math> and <math> y=\frac{-1-i\sqrt{3}}{2} </math>, where <math> i^2=-1 </math>, then which of the following is ''not'' correct?<br />
<br />
<math> \mathrm{(A)\ } x^5+y^5=-1 \qquad \mathrm{(B) \ }x^7+y^7=-1 \qquad \mathrm{(C) \ } x^9+y^9=-1 \qquad </math> <br />
<br />
<math> \mathrm{(D) \ } x^{11}+y^{11}=-1 \qquad \mathrm{(E) \ }x^{13}+y^{13}=-1 </math><br />
<br />
[[1985 AHSME Problems/Problem 23|Solution]]<br />
==Problem 24==<br />
A non-zero [[digit]] is chosen in such a way that the probability of choosing digit <math> d </math> is <math> \log_{10}{(d+1)}-\log_{10}{d} </math>. The probability that the digit <math> 2 </math> is chosen is exactly <math> \frac{1}{2} </math> the probability that the digit is chosen in the set<br />
<br />
<math> \mathrm{(A)\ } \{2, 3\} \qquad \mathrm{(B) \ }\{3, 4\} \qquad \mathrm{(C) \ } \{4, 5, 6, 7, 8\} \qquad \mathrm{(D) \ } \{5, 6, 7, 8, 9\} \qquad \mathrm{(E) \ }\{4, 5, 6, 7, 8, 9\} </math><br />
<br />
[[1985 AHSME Problems/Problem 24|Solution]]<br />
==Problem 25==<br />
The [[volume]] of a certain rectangular solid is <math> 8 \text{cm}^3 </math>, its total [[surface area]] is <math> 32 \text{cm}^2 </math>, and its three dimensions are in [[geometric progression]]. The sums of the lengths in cm of all the edges of this solid is<br />
<br />
<math> \mathrm{(A)\ } 28 \qquad \mathrm{(B) \ }32 \qquad \mathrm{(C) \ } 36 \qquad \mathrm{(D) \ } 40 \qquad \mathrm{(E) \ }44 </math><br />
<br />
[[1985 AHSME Problems/Problem 25|Solution]]<br />
==Problem 26==<br />
Find the least [[positive integer]] <math> n </math> for which <math> \frac{n-13}{5n+6} </math> is a non-zero reducible fraction.<br />
<br />
<math> \mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \ } 155 \qquad \mathrm{(D) \ } 226 \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 26|Solution]]<br />
==Problem 27==<br />
Consider a sequence <math> x_1, x_2, x_3, \cdots </math> defined by<br />
<br />
<math> x_1=\sqrt[3]{3} </math><br />
<br />
<math> x_2=\sqrt[3]{3}^\sqrt[3]{3} </math><br />
<br />
and in general<br />
<br />
<math> x_n=(x_{n-1})^\sqrt[3]{3} </math> for <math> n>1 </math>.<br />
<br />
What is the smallest value of <math> n </math> for which <math> x_n </math> is an [[integer]]?<br />
<br />
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }27 </math><br />
<br />
[[1985 AHSME Problems/Problem 27|Solution]]<br />
==Problem 28==<br />
In <math> \triangle ABC </math>, we have <math> \angle C=3\angle A, a=27, </math> and <math> c=48 </math>. What is <math> b </math>?<br />
<br />
<asy><br />
defaultpen(linewidth(0.7)+fontsize(10));<br />
pair A=origin, B=(14,0), C=(10,6);<br />
draw(A--B--C--cycle);<br />
label("$A$", A, SW);<br />
label("$B$", B, SE);<br />
label("$C$", C, N);<br />
label("$a$", B--C, dir(B--C)*dir(-90));<br />
label("$b$", A--C, dir(C--A)*dir(-90));<br />
label("$c$", A--B, dir(A--B)*dir(-90));</asy><br />
<br />
<math> \mathrm{(A)\ } 33 \qquad \mathrm{(B) \ }35 \qquad \mathrm{(C) \ } 37 \qquad \mathrm{(D) \ } 39 \qquad \mathrm{(E) \ }\text{not uniquely determined} </math><br />
<br />
[[1985 AHSME Problems/Problem 28|Solution]]<br />
==Problem 29==<br />
In their base <math> 10 </math> representation, the integer <math> a </math> consists of a sequence of <math> 1985 </math> eights and the integer <math> b </math> consists of a sequence of <math> 1985 </math> fives. What is the sum of the digits of the base <math> 10 </math> representation of <math> 9ab </math>?<br />
<br />
<math> \mathrm{(A)\ } 15880 \qquad \mathrm{(B) \ }17856 \qquad \mathrm{(C) \ } 17865 \qquad \mathrm{(D) \ } 17874 \qquad \mathrm{(E) \ }19851 </math><br />
<br />
[[1985 AHSME Problems/Problem 29|Solution]]<br />
<br />
==Problem 30==<br />
Let <math> \lfloor x \rfloor </math> be the greatest integer less than or equal to <math> x </math>. Then the number of real solutions to <math> 4x^2-40\lfloor x \rfloor -51=0 </math> is<br />
<br />
<math> \mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 3 \qquad \mathrm{(E) \ }4 </math><br />
<br />
[[1985 AHSME Problems/Problem 30|Solution]]<br />
==See Also==<br />
*[[AHSME]]<br />
*[[1985 AHSME]]</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems/Problem_9&diff=445861985 AHSME Problems/Problem 92012-02-09T03:29:21Z<p>Freddylukai: /* Solution */ changed solution</p>
<hr />
<div>==Problem==<br />
The odd positive integers <math> 1, 3, 5, 7, \cdots </math>, are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which <math> 1985 </math> appears in is the<br />
<br />
<asy><br />
int i,j;<br />
for(i=0; i<4; i=i+1) {<br />
label(string(16*i+1), (2*1,-2*i));<br />
label(string(16*i+3), (2*2,-2*i));<br />
label(string(16*i+5), (2*3,-2*i));<br />
label(string(16*i+7), (2*4,-2*i));<br />
}<br />
for(i=0; i<3; i=i+1) {<br />
for(j=0; j<4; j=j+1) {<br />
label(string(16*i+15-2*j), (2*j,-2*i-1));<br />
}}<br />
dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10));<br />
for(i=-10; i<-6; i=i+1) {<br />
for(j=1; j<4; j=j+1) {<br />
dot((2*j,i));<br />
}}</asy><br />
<br />
<math> \mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \ } \text{third} \qquad \mathrm{(D) \ } \text{fourth} \qquad \mathrm{(E) \ }\text{fifth} </math><br />
<br />
==Solution==<br />
<math>\text{Let us take each number mod 16. Then we have the following pattern:}</math><br />
<br />
<math>\text{ 1 3 5 7}</math><br />
<br />
<math>\text{15 13 11 9 }</math><br />
<br />
<math>\text{ 1 3 5 7}</math><br />
<br />
<math>\text{We can clearly see that all terms congruent to 1 mod 16 will appear in the second column. Since we can see that 1985}\equiv</math> <math>\text{1 (mod 16) , 1985 must appear in the second column.}</math><br />
<br />
<math>\text{Thus, the answer is } \fbox{(B)}</math><br />
<br />
==See Also==<br />
{{AHSME box|year=1985|num-b=8|num-a=10}}</div>Freddylukaihttps://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems&diff=445831985 AHSME Problems2012-02-09T03:11:34Z<p>Freddylukai: /* Problem 9 */</p>
<hr />
<div>==Problem 1==<br />
If <math> 2x+1=8 </math>, then <math> 4x+1= </math><br />
<br />
<math> \mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{(D) \ } 18 \qquad \mathrm{(E) \ }19 </math><br />
<br />
[[1985 AHSME Problems/Problem 1|Solution]]<br />
==Problem 2==<br />
In an arcade game, the "monster" is the shaded sector of a [[circle]] of [[radius]] <math> 1 </math> cm, as shown in the figure. The missing piece (the mouth) has central [[angle]] <math> 60^\circ </math>. What is the [[perimeter]] of the monster in cm?<br />
<br />
<asy><br />
size(100);<br />
defaultpen(linewidth(0.7));<br />
filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black);<br />
label("1", (sqrt(3)/4, 1/4), NW);<br />
label("$60^\circ$", (1,0));</asy><br />
<br />
<math> \mathrm{(A)\ } \pi+2 \qquad \mathrm{(B) \ }2\pi \qquad \mathrm{(C) \ } \frac{5}{3}\pi \qquad \mathrm{(D) \ } \frac{5}{6}\pi+2 \qquad \mathrm{(E) \ }\frac{5}{3}\pi+2 </math><br />
<br />
[[1985 AHSME Problems/Problem 2|Solution]]<br />
==Problem 3==<br />
In right <math> \triangle ABC </math> with legs <math> 5 </math> and <math> 12 </math>, arcs of circles are drawn, one with center <math> A </math> and radius <math> 12 </math>, the other with center <math> B </math> and radius <math> 5 </math>. They intersect the [[hypotenuse]] at <math> M </math> and <math> N </math>. Then, <math> MN </math> has length: <br />
<br />
<asy><br />
defaultpen(linewidth(0.7)+fontsize(10));<br />
pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A);<br />
real r=degrees(B);<br />
draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270));<br />
pair point=incenter(A,B,C);<br />
label("$A$", A, dir(point--A));<br />
label("$B$", B, dir(point--B));<br />
label("$C$", C, dir(point--C));<br />
label("$M$", M, dir(point--M));<br />
label("$N$", N, dir(point--N));<br />
label("$12$", (6,0), S);<br />
label("$5$", (12,3.5), E);</asy><br />
<br />
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }\frac{13}{5} \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 4 \qquad \mathrm{(E) \ }\frac{24}{5} </math><br />
<br />
[[1985 AHSME Problems/Problem 3|Solution]]<br />
==Problem 4==<br />
A large bag of coins contains pennies, dimes, and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is<br />
<br />
<math> \mathrm{(A)\ } </math>&#036;<math>306 \qquad \mathrm{(B) \ } </math>&#036;<math>333 \qquad \mathrm{(C)\ } </math>&#036;<math>342 \qquad \mathrm{(D) \ } </math>&#036;<math>348 \qquad \mathrm{(E) \ } </math>&#036;<math>360 </math><br />
<br />
[[1985 AHSME Problems/Problem 4|Solution]]<br />
==Problem 5==<br />
Which terms must be removed from the sum<br />
<br />
<math> \frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12} </math><br />
<br />
if the sum of the remaining terms is equal to <math> 1 </math>?<br />
<br />
<math> \mathrm{(A)\ } \frac{1}{4}\text{ and }\frac{1}{8} \qquad \mathrm{(B) \ }\frac{1}{4}\text{ and }\frac{1}{12} \qquad \mathrm{(C) \ } \frac{1}{8}\text{ and }\frac{1}{12} \qquad \mathrm{(D) \ } \frac{1}{6}\text{ and }\frac{1}{10} \qquad \mathrm{(E) \ }\frac{1}{8}\text{ and }\frac{1}{10} </math><br />
<br />
[[1985 AHSME Problems/Problem 5|Solution]]<br />
==Problem 6==<br />
One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is <math> \frac{2}{3} </math> of the [[probability]] that a girl is chosen. The [[ratio]] of the number of boys to the total number of boys and girls is<br />
<br />
<math> \mathrm{(A)\ } \frac{1}{3} \qquad \mathrm{(B) \ }\frac{2}{5} \qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ } \frac{3}{5} \qquad \mathrm{(E) \ }\frac{2}{3} </math><br />
<br />
[[1985 AHSME Problems/Problem 6|Solution]]<br />
==Problem 7==<br />
In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from left to right. Thus, <math> a\times b-c </math> in such languages means the same as <math> a(b-c) </math> in ordinary algebraic notation. If <math> a\div b-c+d </math> is evaluated in such a language, the result in ordinary algebraic notation would be<br />
<br />
<math> \mathrm{(A)\ } \frac{a}{b}-c+d \qquad \mathrm{(B) \ }\frac{a}{b}-c-d \qquad \mathrm{(C) \ } \frac{d+c-b}{a} \qquad \mathrm{(D) \ } \frac{a}{b-c+d} \qquad \mathrm{(E) \ }\frac{a}{b-c-d} </math><br />
<br />
[[1985 AHSME Problems/Problem 7|Solution]]<br />
==Problem 8==<br />
Let <math> a, a', b, </math> and <math> b' </math> be real numbers with <math> a </math> and <math> a' </math> nonzero. The solution to <math> ax+b=0 </math> is less than the solution to <math> a'x+b'=0 </math> if and only if <br />
<br />
<math> \mathrm{(A)\ } a'b<ab' \qquad \mathrm{(B) \ }ab'<a'b \qquad \mathrm{(C) \ } ab<a'b' \qquad \mathrm{(D) \ } \frac{b}{a}<\frac{b'}{a'} \qquad </math><br />
<br />
<math> \mathrm{(E) \ }\frac{b'}{a'}<\frac{b}{a} </math><br />
<br />
[[1985 AHSME Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
The odd positive integers <math> 1, 3, 5, 7, \cdots </math>, are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which <math> 1985 </math> appears in is the<br />
<br />
<asy><br />
int i,j;<br />
for(i=0; i<4; i=i+1) {<br />
label(string(16*i+1), (2*1,-2*i));<br />
label(string(16*i+3), (2*2,-2*i));<br />
label(string(16*i+5), (2*3,-2*i));<br />
label(string(16*i+7), (2*4,-2*i));<br />
}<br />
for(i=0; i<3; i=i+1) {<br />
for(j=0; j<4; j=j+1) {<br />
label(string(16*i+15-2*j), (2*j,-2*i-1));<br />
}}<br />
dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10));<br />
for(i=-10; i<-6; i=i+1) {<br />
for(j=1; j<4; j=j+1) {<br />
dot((2*j,i));<br />
}}</asy><br />
<br />
<math> \mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \ } \text{third} \qquad \mathrm{(D) \ } \text{fourth} \qquad \mathrm{(E) \ }\text{fifth} </math><br />
<br />
[[1985 AHSME Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
An arbitrary [[circle]] can intersect the [[graph]] <math> y=\sin x </math> in<br />
<br />
<math> \mathrm{(A) } \text{at most }2\text{ points} \qquad \mathrm{(B) }\text{at most }4\text{ points} \qquad \mathrm{(C) } \text{at most }6\text{ points} \qquad \mathrm{(D) } \text{at most }8\text{ points}\qquad \mathrm{(E) }\text{more than }16\text{ points} </math><br />
<br />
[[1985 AHSME Problems/Problem 10|Solution]]<br />
==Problem 11==<br />
How many '''distinguishable''' rearrangements of the letters in CONTEST have both the vowels first? (For instance, OETCNST is one such arrangement but OTETSNC is not.)<br />
<br />
<math> \mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 720 \qquad \mathrm{(E) \ }2520 </math><br />
<br />
[[1985 AHSME Problems/Problem 11|Solution]]<br />
==Problem 12==<br />
Let's write <math> p, q, </math> and <math> r </math> as three distinct [[prime number]]s, where <math> 1 </math> is not a prime. Which of the following is the smallest positive [[perfect cube]] leaving <math> n=pq^2r^4 </math> as a [[divisor]]?<br />
<br />
<math> \mathrm{(A)\ } p^8q^8r^8 \qquad \mathrm{(B) \ }(pq^2r^2)^3 \qquad \mathrm{(C) \ } (p^2q^2r^2)^3 \qquad \mathrm{(D) \ } (pqr^2)^3 \qquad \mathrm{(E) \ }4p^3q^3r^3 </math><br />
<br />
[[1985 AHSME Problems/Problem 12|Solution]]<br />
==Problem 13==<br />
Pegs are put in a board <math> 1 </math> unit apart both horizontally and vertically. A rubber band is stretched over <math> 4 </math> pegs as shown in the figure, forming a [[quadrilateral]]. Its [[area]] in square units is<br />
<br />
<asy><br />
int i,j;<br />
for(i=0; i<5; i=i+1) {<br />
for(j=0; j<4; j=j+1) {<br />
dot((i,j));<br />
}}<br />
draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7));</asy><br />
<br />
<math> \mathrm{(A)\ } 4 \qquad \mathrm{(B) \ }4.5 \qquad \mathrm{(C) \ } 5 \qquad \mathrm{(D) \ } 5.5 \qquad \mathrm{(E) \ }6 </math><br />
<br />
[[1985 AHSME Problems/Problem 13|Solution]]<br />
==Problem 14==<br />
Exactly three of the interior angles of a convex [[polygon]] are obtuse. What is the maximum number of sides of such a polygon?<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 />
[[1985 AHSME Problems/Problem 14|Solution]]<br />
==Problem 15==<br />
If <math> a </math> and <math> b </math> are positive numbers such that <math> a^b=b^a </math> and <math> b=9a </math>, then the value of <math> a </math> is:<br />
<br />
<math> \mathrm{(A)\ } 9 \qquad \mathrm{(B) \ }\frac{1}{9} \qquad \mathrm{(C) \ } \sqrt[9]{9} \qquad \mathrm{(D) \ } \sqrt[3]{9} \qquad \mathrm{(E) \ }\sqrt[4]{3} </math><br />
<br />
[[1985 AHSME Problems/Problem 15|Solution]]<br />
==Problem 16==<br />
If <math> A=20^\circ </math> and <math> B=25^\circ </math>, then the value of <math> (1+\tan A)(1+\tan B) </math> is<br />
<br />
<math> \mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2(\tan A+\tan B) \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 16|Solution]]<br />
==Problem 17==<br />
[[Diagonal]] <math> DB </math> of [[rectangle]] <math> ABCD </math> is divided into <math> 3 </math> segments of length <math> 1 </math> by [[parallel]] lines <math> L </math> and <math> L' </math> that pass through <math> A </math> and <math> C </math> and are [[perpendicular]] to <math> DB </math>. The area of <math> ABCD </math>, rounded to the nearest tenth, is <br />
<br />
<asy><br />
defaultpen(linewidth(0.7)+fontsize(10));<br />
real x=sqrt(6), y=sqrt(3), a=0.4;<br />
pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D);<br />
real r=degrees(B);<br />
pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90);<br />
markscalefactor=0.02;<br />
draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B)^^rightanglemark(N1,E,B));<br />
pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135);<br />
label("A", A, NE);<br />
label("B", B, NE);<br />
label("C", C, dir(0));<br />
label("D", D, dir(180));<br />
label("$L$", (x/2,0), SW);<br />
label("$L^\prime$", C, SW);<br />
label("1", D--F, NW);<br />
label("1", F--E, SE);<br />
label("1", E--B, SE);<br />
clip(W--X--Y--Z--cycle);</asy><br />
<br />
<math> \mathrm{(A)\ } 4.1 \qquad \mathrm{(B) \ }4.2 \qquad \mathrm{(C) \ } 4.3 \qquad \mathrm{(D) \ } 4.4 \qquad \mathrm{(E) \ }4.5 </math><br />
<br />
[[1985 AHSME Problems/Problem 17|Solution]]<br />
==Problem 18==<br />
Six bags of marbles contain <math> 18, 19, 21, 23, 25, </math> and <math> 34 </math> marbles, respectively. One bag contains chipped marbles only. The other <math> 5 </math> bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?<br />
<br />
<math> \mathrm{(A)\ } 18 \qquad \mathrm{(B) \ }19 \qquad \mathrm{(C) \ } 21 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ }25 </math><br />
<br />
[[1985 AHSME Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
Consider the graphs <math> y=Ax^2 </math> and <math> y^2+3=x^2+4y </math>, where <math> A </math> is a positive constant and <math> x </math> and <math> y </math> are real variables. In how many points do the two graphs intersect?<br />
<br />
<math> \mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad </math> <br />
<br />
<math> \mathrm{(C) \ }\text{at least }1,\text{ but the number varies for different positive values of }A \qquad </math> <br />
<br />
<math> \mathrm{(D) \ }0\text{ for at least one positive value of }A \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 19|Solution]]<br />
==Problem 20==<br />
A wooden [[cube]] with edge length <math> n </math> units (where <math> n </math> is an integer <math> >2 </math>) is painted black all over. By slices parallel to its faces, the cube is cut into <math> n^3 </math> smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is <math> n </math>?<br />
<br />
<math> \mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 8 \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 20|Solution]]<br />
==Problem 21==<br />
How many integers <math> x </math> satisfy the equation <math> (x^2-x-1)^{x+2}=1 </math><br />
<br />
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 21|Solution]]<br />
==Problem 22==<br />
In a circle with center <math> O </math>, <math> AD </math> is a [[diameter]], <math> ABC </math> is a [[chord]], <math> BO=5 </math>, and <math> \angle ABO=\stackrel{\frown}{CD}=60^\circ </math>. Then the length of <math> BC </math> is:<br />
<br />
<asy><br />
defaultpen(linewidth(0.7)+fontsize(10));<br />
pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C);<br />
draw(C--A--D^^B--O^^Circle(O,1));<br />
pair point=O;<br />
label("$A$", A, dir(point--A));<br />
label("$B$", B, dir(point--B));<br />
label("$C$", C, dir(point--C));<br />
label("$D$", D, dir(point--D));<br />
label("$O$", O, dir(305));<br />
label("$5$", B--O, dir(O--B)*dir(90));<br />
label("$60^\circ$", dir(185), dir(185));<br />
label("$60^\circ$", B+0.05*dir(-25), dir(-25));</asy><br />
<br />
<math> \mathrm{(A)\ } 3 \qquad \mathrm{(B) \ }3+\sqrt{3} \qquad \mathrm{(C) \ } 5-\frac{\sqrt{3}}{2} \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ }\text{none of the above} </math><br />
<br />
[[1985 AHSME Problems/Problem 22|Solution]]<br />
==Problem 23==<br />
If <math> x=\frac{-1+i\sqrt{3}}{2} </math> and <math> y=\frac{-1-i\sqrt{3}}{2} </math>, where <math> i^2=-1 </math>, then which of the following is ''not'' correct?<br />
<br />
<math> \mathrm{(A)\ } x^5+y^5=-1 \qquad \mathrm{(B) \ }x^7+y^7=-1 \qquad \mathrm{(C) \ } x^9+y^9=-1 \qquad </math> <br />
<br />
<math> \mathrm{(D) \ } x^{11}+y^{11}=-1 \qquad \mathrm{(E) \ }x^{13}+y^{13}=-1 </math><br />
<br />
[[1985 AHSME Problems/Problem 23|Solution]]<br />
==Problem 24==<br />
A non-zero [[digit]] is chosen in such a way that the probability of choosing digit <math> d </math> is <math> \log_{10}{(d+1)}-\log_{10}{d} </math>. The probability that the digit <math> 2 </math> is chosen is exactly <math> \frac{1}{2} </math> the probability that the digit is chosen in the set<br />
<br />
<math> \mathrm{(A)\ } \{2, 3\} \qquad \mathrm{(B) \ }\{3, 4\} \qquad \mathrm{(C) \ } \{4, 5, 6, 7, 8\} \qquad \mathrm{(D) \ } \{5, 6, 7, 8, 9\} \qquad \mathrm{(E) \ }\{4, 5, 6, 7, 8, 9\} </math><br />
<br />
[[1985 AHSME Problems/Problem 24|Solution]]<br />
==Problem 25==<br />
The [[volume]] of a certain rectangular solid is <math> 8 \text{cm}^3 </math>, its total [[surface area]] is <math> 32 \text{cm}^2 </math>, and its three dimensions are in [[geometric progression]]. The sums of the lengths in cm of all the edges of this solid is<br />
<br />
<math> \mathrm{(A)\ } 28 \qquad \mathrm{(B) \ }32 \qquad \mathrm{(C) \ } 36 \qquad \mathrm{(D) \ } 40 \qquad \mathrm{(E) \ }44 </math><br />
<br />
[[1985 AHSME Problems/Problem 25|Solution]]<br />
==Problem 26==<br />
Find the least [[positive integer]] <math> n </math> for which <math> \frac{n-13}{5n+6} </math> is a non-zero reducible fraction.<br />
<br />
<math> \mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \ } 155 \qquad \mathrm{(D) \ } 226 \qquad \mathrm{(E) \ }\text{none of these} </math><br />
<br />
[[1985 AHSME Problems/Problem 26|Solution]]<br />
==Problem 27==<br />
Consider a sequence <math> x_1, x_2, x_3, \cdots </math> defined by<br />
<br />
<math> x_1=\sqrt[3]{3} </math><br />
<br />
<math> x_2=\sqrt[3]{3}^\sqrt[3]{3} </math><br />
<br />
and in general<br />
<br />
<math> x_n=(x_{n-1})^\sqrt[3]{3} </math> for <math> n>1 </math>.<br />
<br />
What is the smallest value of <math> n </math> for which <math> x_n </math> is an [[integer]]?<br />
<br />
<math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }27 </math><br />
<br />
[[1985 AHSME Problems/Problem 27|Solution]]<br />
==Problem 28==<br />
In <math> \triangle ABC </math>, we have <math> \angle C=3\angle A, a=27, </math> and <math> c=48 </math>. What is <math> b </math>?<br />
<br />
<asy><br />
defaultpen(linewidth(0.7)+fontsize(10));<br />
pair A=origin, B=(14,0), C=(10,6);<br />
draw(A--B--C--cycle);<br />
label("$A$", A, SW);<br />
label("$B$", B, SE);<br />
label("$C$", C, N);<br />
label("$a$", B--C, dir(B--C)*dir(-90));<br />
label("$b$", A--C, dir(C--A)*dir(-90));<br />
label("$c$", A--B, dir(A--B)*dir(-90));</asy><br />
<br />
<math> \mathrm{(A)\ } 33 \qquad \mathrm{(B) \ }35 \qquad \mathrm{(C) \ } 37 \qquad \mathrm{(D) \ } 39 \qquad \mathrm{(E) \ }\text{not uniquely determined} </math><br />
<br />
[[1985 AHSME Problems/Problem 28|Solution]]<br />
==Problem 29==<br />
In their base <math> 10 </math> representation, the integer <math> a </math> consists of a sequence of <math> 1985 </math> eights and the integer <math> b </math> consists of a sequence of <math> 1985 </math> fives. What is the sum of the digits of the base <math> 10 </math> representation of <math> 9ab </math>?<br />
<br />
<math> \mathrm{(A)\ } 15880 \qquad \mathrm{(B) \ }17856 \qquad \mathrm{(C) \ } 17865 \qquad \mathrm{(D) \ } 17874 \qquad \mathrm{(E) \ }19851 </math><br />
<br />
[[1985 AHSME Problems/Problem 29|Solution]]<br />
<br />
==Problem 30==<br />
Let <math> \lfloor x \rfloor </math> be the greatest integer less than or equal to <math> x </math>. Then the number of real solutions to <math> 4x^2-40\lfloor x \rfloor -51=0 </math> is<br />
<br />
<math> \mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 3 \qquad \mathrm{(E) \ }4 </math><br />
<br />
[[1985 AHSME Problems/Problem 30|Solution]]<br />
==See Also==<br />
*[[AHSME]]<br />
*[[1985 AHSME]]</div>Freddylukai