Difference between revisions of "1996 AHSME Problems"

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{{AHSME Problems
 +
|year = 1996
 +
}}
 
==Problem 1==
 
==Problem 1==
  
 
The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?
 
The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?
 
   
 
   
<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>
+
<math> \begin{tabular}{rr}&\ \texttt{6 4 1}\\ &\texttt{8 5 2}\\ +&\texttt{9 7 3}\\ \hline  &\texttt{2 4 5 6}\end{tabular} </math>
  
  
Line 20: Line 23:
  
 
==Problem 3==
 
==Problem 3==
 +
 +
<math> \frac{(3!)!}{3!}= </math>
 +
 +
<math> \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 6\qquad\text{(D)}\ 40\qquad\text{(E)}\ 120 </math>
  
 
[[1996 AHSME Problems/Problem 3|Solution]]
 
[[1996 AHSME Problems/Problem 3|Solution]]
Line 25: Line 32:
 
==Problem 4==
 
==Problem 4==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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
 +
 
 +
<math> \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
  
[[1996 AHSME Problems/Problem |Solution]]
+
Given that <math> 0 < a < b < c < d </math>, which of the following is the largest?
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
  
[[1996 AHSME Problems/Problem |Solution]]
+
If <math> f(x) = x^{(x+1)}(x+2)^{(x+3)} </math>, then <math> f(0)+f(-1)+f(-2)+f(-3) = </math>
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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?
 +
 
 +
<math> \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
  
[[1996 AHSME Problems/Problem |Solution]]
+
If <math>3 = k\cdot 2^r</math> and <math>15 = k\cdot 4^r</math>, then <math>r = </math>
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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>?
 +
<asy>
 +
real r=sqrt(2)/2;
 +
draw(origin--(8,0)--(8,-1)--(0,-1)--cycle);
 +
draw(origin--(8,0)--(8+r, r)--(r,r)--cycle);
 +
filldraw(origin--(-6*r, -6*r)--(8-6*r, -6*r)--(8, 0)--cycle, white, black);
 +
filldraw(origin--(8,0)--(8,6)--(0,6)--cycle, white, black);
 +
pair A=(6,0), B=(2,0), C=(2,4), D=(6,4), P=B+1*dir(-65);
 +
draw(A--P--B--C--D--cycle);
 +
dot(A^^B^^C^^D^^P);
 +
label("$A$", A, dir((4,2)--A));
 +
label("$B$", B, dir((4,2)--B));
 +
label("$C$", C, dir((4,2)--C));
 +
label("$D$", D, dir((4,2)--D));
 +
label("$P$", P, dir((4,2)--P));</asy>
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
  
[[1996 AHSME Problems/Problem |Solution]]
+
How many line segments have both their endpoints located at the vertices of a given cube?
 +
 
 +
<math> \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 24\qquad\text{(D)}\ 28\qquad\text{(E)}\ 56 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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.
 +
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 11|Solution]]
  
 
==Problem 12==
 
==Problem 12==
  
[[1996 AHSME Problems/Problem |Solution]]
+
A function <math>f</math> from the integers to the integers is defined as follows:
 +
 
 +
<cmath> f(x) =\begin{cases}n+3 &\text{if n is odd}\\ n/2 &\text{if n is even}\end{cases} </cmath>
 +
 
 +
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>?
 +
 
 +
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15 </math>
 +
 
 +
 
 +
[[1996 AHSME Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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?
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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>
 +
 
 +
<math> \text{(A)}\ 200\qquad\text{(B)}\ 360\qquad\text{(C)}\ 400\qquad\text{(D)}\ 900\qquad\text{(E)}\ 2250 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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>?
 +
 
 +
<asy>
 +
int i;
 +
for(i=0; i<8; i=i+1) {
 +
dot((i,0)^^(i,5));
 +
}
 +
for(i=1; i<5; i=i+1) {
 +
dot((0,i)^^(7,i));
 +
}
 +
draw(origin--(7,0)--(7,5)--(0,5)--cycle, linewidth(0.8));
 +
pair P=(3.5, 2.5);
 +
draw((0,4)--P--(0,3)^^(2,0)--P--(3,0));
 +
label("$B$", (2.3,0), NE);
 +
label("$A$", (0,3.7), SE);
 +
</asy>
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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?
 +
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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>?
 +
<asy>
 +
pair A=origin, B=(10,0), C=(10,7), D=(0,7), E=(5,0), F=(0,2);
 +
draw(A--B--C--D--cycle, linewidth(0.8));
 +
draw(E--C--F);
 +
dot(A^^B^^C^^D^^E^^F);
 +
label("$A$", A, dir((5, 3.5)--A));
 +
label("$B$", B, dir((5, 3.5)--B));
 +
label("$C$", C, dir((5, 3.5)--C));
 +
label("$D$", D, dir((5, 3.5)--D));
 +
label("$E$", E, dir((5, 3.5)--E));
 +
label("$F$", F, dir((5, 3.5)--F));
 +
label("$2$", (0,1), dir(0));
 +
label("$6$", (7.5,0), N);</asy>
 +
<math> \text{(A)}\ 110\qquad\text{(B)}\ 120\qquad\text{(C)}\ 130\qquad\text{(D)}\ 140\qquad\text{(E)}\ 150 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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?
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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?
 +
 
 +
<asy>
 +
size(120);
 +
draw(rotate(30)*polygon(6));
 +
draw(scale(2/sqrt(3))*polygon(6));
 +
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);
 +
dot(A^^B^^C^^D^^E^^F);
 +
label("$A$", A, dir(origin--A));
 +
label("$B$", B, dir(origin--B));
 +
label("$C$", C, dir(origin--C));
 +
label("$D$", D, dir(origin--D));
 +
label("$E$", E, dir(origin--E));
 +
label("$F$", F, dir(origin--F));
 +
</asy>
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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>?
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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
 +
 +
<asy>
 +
size(120);
 +
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);
 +
dot(A^^B^^C^^D^^E);
 +
draw(A--D--B--A--C--B);
 +
markscalefactor=0.005;
 +
draw(rightanglemark(A, E, B));
 +
dot(A^^B^^C^^D^^E);
 +
pair point=midpoint(A--M);
 +
label("$A$", A, dir(point--A));
 +
label("$B$", B, dir(point--B));
 +
label("$C$", C, dir(point--C));
 +
label("$D$", D, dir(point--D));
 +
label("$E$", E, dir(point--E));
 +
</asy>
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 22==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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
 +
evenly spaced around a circle. All quadruples are equally likely to be chosen.
 +
What is the probability that the chord <math>AB</math> intersects the chord <math>CD</math>?
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 22|Solution]]
  
 
==Problem 23==
 
==Problem 23==
  
[[1996 AHSME Problems/Problem |Solution]]
+
The sum of the lengths of the twelve edges of a rectangular box is <math>140</math>, and
 +
the distance from one corner of the box to the farthest corner is <math>21</math>. The total
 +
surface area of the box is
 +
 
 +
<math> \text{(A)}\ 776\qquad\text{(B)}\ 784\qquad\text{(C)}\ 798\qquad\text{(D)}\ 800\qquad\text{(E)}\ 812 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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
 +
 
 +
<math> \text{(A)}\ 1996\qquad\text{(B)}\ 2419\qquad\text{(C)}\ 2429\qquad\text{(D)}\ 2439\qquad\text{(E)}\ 2449 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
  
[[1996 AHSME Problems/Problem |Solution]]
+
Given that <math>x^2 + y^2 = 14x + 6y + 6</math>, what is the largest possible value that <math>3x + 4y</math> can have?
 +
 
 +
<math> \text{(A)}\ 72\qquad\text{(B)}\ 73\qquad\text{(C)}\ 74\qquad\text{(D)}\ 75\qquad\text{(E)}\ 76 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 25|Solution]]
  
 
==Problem 26==
 
==Problem 26==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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:
 +
 
 +
(a) the selection of four red marbles;
 +
 
 +
(b) the selection of one white and three red marbles;
 +
 
 +
(c) the selection of one white, one blue, and two red marbles; and
 +
 
 +
(d) the selection of one marble of each color.
 +
 
 +
What is the smallest number of marbles satisfying the given condition?
 +
 
 +
<math> \text{(A)}\ 19\qquad\text{(B)}\ 21\qquad\text{(C)}\ 46\qquad\text{(D)}\ 69\qquad\text{(E)}\ \text{more than 69} </math>
 +
 
 +
[[1996 AHSME Problems/Problem 26|Solution]]
  
 
==Problem 27==
 
==Problem 27==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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?
 +
 
 +
<math> \text{(A)}\ 7\qquad\text{(B)}\ 9\qquad\text{(C)}\ 11\qquad\text{(D)}\ 13\qquad\text{(E)}\ 15 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 27|Solution]]
  
 
==Problem 28==
 
==Problem 28==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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
 +
<math>A</math>, <math>B</math>, and <math>C</math> is closest to
 +
 
 +
<asy>
 +
size(120);
 +
import three;
 +
currentprojection=orthographic(1, 4/5, 1/3);
 +
draw(box(O, (4,4,3)));
 +
triple A=(0,4,3), B=(0,0,0) , C=(4,4,0), D=(0,4,0);
 +
draw(A--B--C--cycle, linewidth(0.9));
 +
label("$A$", A, NE);
 +
label("$B$", B, NW);
 +
label("$C$", C, S);
 +
label("$D$", D, E);
 +
label("$4$", (4,2,0), SW);
 +
label("$4$", (2,4,0), SE);
 +
label("$3$", (0, 4, 1.5), E);
 +
</asy>
 +
 
 +
<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>
 +
 
 +
[[1996 AHSME Problems/Problem 28|Solution]]
  
 
==Problem 29==
 
==Problem 29==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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?
 +
 
 +
<math> \text{(A)}\ 32\qquad\text{(B)}\ 34\qquad\text{(C)}\ 35\qquad\text{(D)}\ 36\qquad\text{(E)}\ 38 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 29|Solution]]
  
 
==Problem 30==
 
==Problem 30==
  
[[1996 AHSME Problems/Problem |Solution]]
+
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>.
 +
 +
<math> \text{(A)}\ 309\qquad\text{(B)}\ 349\qquad\text{(C)}\ 369\qquad\text{(D)}\ 389\qquad\text{(E)}\ 409 </math>
 +
 
 +
[[1996 AHSME Problems/Problem 30|Solution]]
 +
 
 +
== See also ==
 +
 
 +
* [[AMC 12 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
 +
 
 +
{{AHSME box|year=1996|before=[[1995 AHSME]]|after=[[1997 AHSME]]}} 
 +
 
 +
{{MAA Notice}}

Latest revision as of 13:39, 19 February 2020

1996 AHSME (Answer Key)
Printable versions: WikiAoPS ResourcesPDF

Instructions

  1. This is a 30-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers.
  4. Figures are not necessarily drawn to scale.
  5. You will have 90 minutes working time to complete the test.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Problem 1

The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?

$\begin{tabular}{rr}&\ \texttt{6 4 1}\\ &\texttt{8 5 2}\\ +&\texttt{9 7 3}\\ \hline  &\texttt{2 4 5 6}\end{tabular}$


$\text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E)}\ 8$


Solution

Problem 2

Each day Walter gets $3$ dollars for doing his chores or $5$ dollars for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $36$ dollars. On how many days did Walter do them exceptionally well?

$\text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 7$

Solution

Problem 3

$\frac{(3!)!}{3!}=$

$\text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 6\qquad\text{(D)}\ 40\qquad\text{(E)}\ 120$

Solution

Problem 4

Six numbers from a list of nine integers are $7,8,3,5, 9$ and $5$. The largest possible value of the median of all nine numbers in this list is

$\text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9$

Solution

Problem 5

Given that $0 < a < b < c < d$, which of the following is the largest?

$\text{(A)}\  \frac{a+b}{c+d} \qquad\text{(B)}\ \frac{a+d}{b+c} \qquad\text{(C)}\  \frac{b+c}{a+d} \qquad\text{(D)}\  \frac{b+d}{a+c} \qquad\text{(E)}\ \frac{c+d}{a+b}$

Solution

Problem 6

If $f(x) = x^{(x+1)}(x+2)^{(x+3)}$, then $f(0)+f(-1)+f(-2)+f(-3) =$

$\text{(A)}\ -\frac{8}{9}\qquad\text{(B)}\ 0\qquad\text{(C)}\ \frac{8}{9}\qquad\text{(D)}\ 1\qquad\text{(E)}\ \frac{10}{9}$

Solution

Problem 7

A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for the father and $0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is $9.45$, which of the following could be the age of the youngest child?

$\text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5$

Solution

Problem 8

If $3 = k\cdot 2^r$ and $15 = k\cdot 4^r$, then $r =$

$\text{(A)}\ -\log_{2}5\qquad\text{(B)}\ \log_{5}2\qquad\text{(C)}\ \log_{10}5\qquad\text{(D)}\ \log_{2}5\qquad\text{(E)}\ \frac{5}{2}$

Solution

Problem 9

Triangle $PAB$ and square $ABCD$ are in perpendicular planes. Given that $PA = 3, PB = 4$ and $AB = 5$, what is $PD$? [asy] real r=sqrt(2)/2; draw(origin--(8,0)--(8,-1)--(0,-1)--cycle); draw(origin--(8,0)--(8+r, r)--(r,r)--cycle); filldraw(origin--(-6*r, -6*r)--(8-6*r, -6*r)--(8, 0)--cycle, white, black); filldraw(origin--(8,0)--(8,6)--(0,6)--cycle, white, black); pair A=(6,0), B=(2,0), C=(2,4), D=(6,4), P=B+1*dir(-65); draw(A--P--B--C--D--cycle); dot(A^^B^^C^^D^^P); label("$A$", A, dir((4,2)--A)); label("$B$", B, dir((4,2)--B)); label("$C$", C, dir((4,2)--C)); label("$D$", D, dir((4,2)--D)); label("$P$", P, dir((4,2)--P));[/asy]

$\text{(A)}\ 5\qquad\text{(B)}\ \sqrt{34} \qquad\text{(C)}\ \sqrt{41}\qquad\text{(D)}\ 2\sqrt{13}\qquad\text{(E)}\ 8$

Solution

Problem 10

How many line segments have both their endpoints located at the vertices of a given cube?

$\text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 24\qquad\text{(D)}\ 28\qquad\text{(E)}\ 56$

Solution

Problem 11

Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.

$\text{(A)}\ \frac{\pi} 4\qquad\text{(B)}\ 4-\pi\qquad\text{(C)}\ \frac{\pi} 2\qquad\text{(D)}\ \pi\qquad\text{(E)}\ 2\pi$

Solution

Problem 12

A function $f$ from the integers to the integers is defined as follows:

\[f(x) =\begin{cases}n+3 &\text{if n is odd}\\ n/2 &\text{if n is even}\end{cases}\]

Suppose $k$ is odd and $f(f(f(k))) = 27$. What is the sum of the digits of $k$?

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15$


Solution

Problem 13

Sunny runs at a steady rate, and Moonbeam runs $m$ times as fast, where $m$ is a number greater than 1. If Moonbeam gives Sunny a head start of $h$ meters, how many meters must Moonbeam run to overtake Sunny?

$\text{(A)}\ hm\qquad\text{(B)}\ \frac{h}{h+m}\qquad\text{(C)}\ \frac{h}{m-1}\qquad\text{(D)}\ \frac{hm}{m-1}\qquad\text{(E)}\ \frac{h+m}{m-1}$

Solution

Problem 14

Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1)+E(2)+E(3)+\cdots+E(100)$

$\text{(A)}\ 200\qquad\text{(B)}\ 360\qquad\text{(C)}\ 400\qquad\text{(D)}\ 900\qquad\text{(E)}\ 2250$

Solution

Problem 15

Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n=5, m=7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$?

[asy] int i; for(i=0; i<8; i=i+1) { dot((i,0)^^(i,5)); } for(i=1; i<5; i=i+1) { dot((0,i)^^(7,i)); } draw(origin--(7,0)--(7,5)--(0,5)--cycle, linewidth(0.8)); pair P=(3.5, 2.5); draw((0,4)--P--(0,3)^^(2,0)--P--(3,0)); label("$B$", (2.3,0), NE); label("$A$", (0,3.7), SE); [/asy]

$\text{(A)}\ 1\qquad\text{(B)}\ m/n\qquad\text{(C)}\ n/m\qquad\text{(D)}\ 2m/n\qquad\text{(E)}\ 2n/m$

Solution

Problem 16

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?

$\text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{91}{216}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{8}{15}\qquad\text{(E)}\ \frac{7}{12}$

Solution

Problem 17

In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=6$ and $AF=2$. Which of the following is closest to the area of the rectangle $ABCD$? [asy] pair A=origin, B=(10,0), C=(10,7), D=(0,7), E=(5,0), F=(0,2); draw(A--B--C--D--cycle, linewidth(0.8)); draw(E--C--F); dot(A^^B^^C^^D^^E^^F); label("$A$", A, dir((5, 3.5)--A)); label("$B$", B, dir((5, 3.5)--B)); label("$C$", C, dir((5, 3.5)--C)); label("$D$", D, dir((5, 3.5)--D)); label("$E$", E, dir((5, 3.5)--E)); label("$F$", F, dir((5, 3.5)--F)); label("$2$", (0,1), dir(0)); label("$6$", (7.5,0), N);[/asy] $\text{(A)}\ 110\qquad\text{(B)}\ 120\qquad\text{(C)}\ 130\qquad\text{(D)}\ 140\qquad\text{(E)}\ 150$

Solution

Problem 18

A circle of radius $2$ has center at $(2,0)$. A circle of radius $1$ has center at $(5,0)$. A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the $y$-intercept of the line?

$\text{(A)}\ \sqrt{2}/4\qquad\text{(B)}\ 8/3\qquad\text{(C)}\ 1+\sqrt 3\qquad\text{(D)}\ 2\sqrt 2\qquad\text{(E)}\ 3$

Solution

Problem 19

The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon?

[asy] size(120); draw(rotate(30)*polygon(6)); draw(scale(2/sqrt(3))*polygon(6)); 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); dot(A^^B^^C^^D^^E^^F); label("$A$", A, dir(origin--A)); label("$B$", B, dir(origin--B)); label("$C$", C, dir(origin--C)); label("$D$", D, dir(origin--D)); label("$E$", E, dir(origin--E)); label("$F$", F, dir(origin--F)); [/asy]

$\text{(A)}\ \frac{1}{2}\qquad\text{(B)}\ \frac{\sqrt 3}{3}\qquad\text{(C)}\ \frac{2}{3}\qquad\text{(D)}\ \frac{3}{4}\qquad\text{(E)}\ \frac{\sqrt 3}{2}$

Solution

Problem 20

In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(12,16)$ that does not go inside the circle $(x-6)^{2}+(y-8)^{2}= 25$?

$\text{(A)}\ 10\sqrt 3\qquad\text{(B)}\ 10\sqrt 5\qquad\text{(C)}\ 10\sqrt 3+\frac{ 5\pi}{3}\qquad\text{(D)}\ 40\frac{\sqrt{3}}{3}\qquad\text{(E)}\ 10+5\pi$

Solution

Problem 21

Triangles $ABC$ and $ABD$ are isosceles with $AB=AC=BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then $\angle C+\angle D$ is

[asy] size(120); 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); dot(A^^B^^C^^D^^E); draw(A--D--B--A--C--B); markscalefactor=0.005; draw(rightanglemark(A, E, B)); dot(A^^B^^C^^D^^E); pair point=midpoint(A--M); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); [/asy]

$\text{(A)}\ 115^\circ\qquad\text{(B)}\ 120^\circ\qquad\text{(C)}\ 130^\circ\qquad\text{(D)}\ 135^\circ\qquad\text{(E)}\ \text{not uniquely determined}$

Solution

Problem 22

Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $AB$ intersects the chord $CD$?

$\text{(A)}\ \frac{1}{4}\qquad\text{(B)}\ \frac{1}{3}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{2}{3}\qquad\text{(E)}\ \frac{3}{4}$

Solution

Problem 23

The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. The total surface area of the box is

$\text{(A)}\ 776\qquad\text{(B)}\ 784\qquad\text{(C)}\ 798\qquad\text{(D)}\ 800\qquad\text{(E)}\ 812$

Solution

Problem 24

The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots$ consists of $1$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. The sum of the first $1234$ terms of this sequence is

$\text{(A)}\ 1996\qquad\text{(B)}\ 2419\qquad\text{(C)}\ 2429\qquad\text{(D)}\ 2439\qquad\text{(E)}\ 2449$

Solution

Problem 25

Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have?

$\text{(A)}\ 72\qquad\text{(B)}\ 73\qquad\text{(C)}\ 74\qquad\text{(D)}\ 75\qquad\text{(E)}\ 76$

Solution

Problem 26

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:

(a) the selection of four red marbles;

(b) the selection of one white and three red marbles;

(c) the selection of one white, one blue, and two red marbles; and

(d) the selection of one marble of each color.

What is the smallest number of marbles satisfying the given condition?

$\text{(A)}\ 19\qquad\text{(B)}\ 21\qquad\text{(C)}\ 46\qquad\text{(D)}\ 69\qquad\text{(E)}\ \text{more than 69}$

Solution

Problem 27

Consider two solid spherical balls, one centered at $(0, 0,\frac{21}{2})$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac{9}{2}$. How many points with only integer coordinates (lattice points) are there in the intersection of the balls?

$\text{(A)}\ 7\qquad\text{(B)}\ 9\qquad\text{(C)}\ 11\qquad\text{(D)}\ 13\qquad\text{(E)}\ 15$

Solution

Problem 28

On a $4\times 4\times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to

[asy] size(120); import three; currentprojection=orthographic(1, 4/5, 1/3); draw(box(O, (4,4,3))); triple A=(0,4,3), B=(0,0,0) , C=(4,4,0), D=(0,4,0); draw(A--B--C--cycle, linewidth(0.9)); label("$A$", A, NE); label("$B$", B, NW); label("$C$", C, S); label("$D$", D, E); label("$4$", (4,2,0), SW); label("$4$", (2,4,0), SE); label("$3$", (0, 4, 1.5), E); [/asy]

$\text{(A)}\ 1.6\qquad\text{(B)}\ 1.9\qquad\text{(C)}\ 2.1\qquad\text{(D)}\ 2.7\qquad\text{(E)}\ 2.9$

Solution

Problem 29

If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have?

$\text{(A)}\ 32\qquad\text{(B)}\ 34\qquad\text{(C)}\ 35\qquad\text{(D)}\ 36\qquad\text{(E)}\ 38$

Solution

Problem 30

A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

$\text{(A)}\ 309\qquad\text{(B)}\ 349\qquad\text{(C)}\ 369\qquad\text{(D)}\ 389\qquad\text{(E)}\ 409$

Solution

See also

1996 AHSME (ProblemsAnswer KeyResources)
Preceded by
1995 AHSME
Followed by
1997 AHSME
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
All AHSME Problems and Solutions


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