1988 AHSME Problems

Problem 1

Solution

Problem 2

Triangles and are similar, with corresponding to and to . If , and , then is:

Solution

Problem 3

Four rectangular paper strips of length and width are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?

draw((0,0)--(1,0)--(1,4)--(0,4)--(0,0)--(0,1)--(-1,1)--(-1,2)--(0,2)--(0,4)--(-1,4)--(-1,5)--(1,5)--(1,6)--(0,6));
draw((0,6)--(0,5)--(3,5)--(3,6)--(4,6)--(4,2)--(5,2)--(5,1)--(1,1)--(3,1)--(3,0)--(4,0)--(4,1));
draw((1,4)--(3,4)--(3,2)--(1,2)--(4,2)--(3,2)--(3,6)--(4,6)--(4,5)--(5,5)--(5,4)--(4,4));
<\asy>

$\text{(A)}\ 36 \qquad 
\text{(B)}\ 40 \qquad 
\text{(C)}\ 44 \qquad 
\text{(D)}\ 98 \qquad 
\text{(E)}\ 100 $    
  
[[1988 AHSME Problems/Problem 3|Solution]]

==Problem 4==

The slope of the line $\frac{x}{3} + \frac{y}{2} = 1$ is

$\textbf{(A)}\ -\frac{3}{2}\qquad
\textbf{(B)}\ -\frac{2}{3}\qquad
\textbf{(C)}\ \frac{1}{3}\qquad
\textbf{(D)}\ \frac{2}{3}\qquad
\textbf{(E)}\ \frac{3}{2}$    
  
[[1988 AHSME Problems/Problem 4|Solution]]

==Problem 5==

If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then $c$ is

$\textbf{(A)}\ -5\qquad
\textbf{(B)}\ -3\qquad
\textbf{(C)}\ -1\qquad
\textbf{(D)}\ 3\qquad
\textbf{(E)}\ 5$
  
[[1988 AHSME Problems/Problem 5|Solution]]

==Problem 6==

A figure is an equiangular parallelogram if and only if it is a

$\textbf{(A)}\ \text{rectangle}\qquad
\textbf{(B)}\ \text{regular polygon}\qquad
\textbf{(C)}\ \text{rhombus}\qquad
\textbf{(D)}\ \text{square}\qquad
\textbf{(E)}\ \text{trapezoid}$  

[[1988 AHSME Problems/Problem 6|Solution]]

==Problem 7==

Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$
"chunks" and the channel can transmit $120$ chunks per second.

$\textbf{(A)}\ 0.04 \text{ seconds}\qquad
\textbf{(B)}\ 0.4 \text{ seconds}\qquad
\textbf{(C)}\ 4 \text{ seconds}\qquad
\textbf{(D)}\ 4\text{ minutes}\qquad
\textbf{(E)}\ 4\text{ hours}$
  
[[1988 AHSME Problems/Problem 7|Solution]]

==Problem 8==

If $\frac{b}{a} = 2$ and $\frac{c}{b} = 3$, what is the ratio of $a + b$ to $b + c$?

$\textbf{(A)}\ \frac{1}{3}\qquad
\textbf{(B)}\ \frac{3}{8}\qquad
\textbf{(C)}\ \frac{3}{5}\qquad
\textbf{(D)}\ \frac{2}{3}\qquad
\textbf{(E)}\ \frac{3}{4}  $   
  
[[1988 AHSME Problems/Problem 8|Solution]]

==Problem 9==

An $8$'\text{ X }$10$' table sits in the corner of a square room, as in Figure $1$ below. 
The owners desire to move the table to the position shown in Figure $2$. 
The side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart?

defaultpen(linewidth(0.7)+fontsize(10));pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16)...

$\textbf{(A)}\ 11\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 13\qquad
\textbf{(D)}\ 14\qquad
\textbf{(E)}\ 15   $  
  
[[1988 AHSME Problems/Problem 9|Solution]]

==Problem 10==

In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\pm 0.00312$. 
The experimenter wishes to announce a value for $C$ in which every digit is significant. 
That is, whatever $C$ is, the announced value must be the correct result when $C$ is rounded to that number of digits. 
The most accurate value the experimenter can announce for $C$ is

$\textbf{(A)}\ 2\qquad
\textbf{(B)}\ 2.4\qquad
\textbf{(C)}\ 2.43\qquad
\textbf{(D)}\ 2.44\qquad
\textbf{(E)}\ 2.439   $  
  
[[1988 AHSME Problems/Problem 10|Solution]]

==Problem 11==

On each horizontal line in the figure below, the five large dots indicate the populations of cities $A, B, C, D$ and $E$ in the year indicated. 
Which city had the greatest percentage increase in population from $1970$ to $1980$?

defaultpen(linewidth(0.7)+fontsize(10));pair A=(5,0), B=(7,0), C=(10,0), D=(13,0), E=(16,0);pair F=(4,3), G=(5,3), H=(7,3), I...

$\textbf{(A)}\ A\qquad
\textbf{(B)}\ B\qquad
\textbf{(C)}\ C\qquad
\textbf{(D)}\ D\qquad
\textbf{(E)}\ E     $
  
[[1988 AHSME Problems/Problem 11|Solution]]

==Problem 12==

Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. 
Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. 
Which digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer?

$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 8\qquad
\textbf{(D)}\ 9\qquad
\textbf{(E)}\ \text{each digit is equally likely}     $
  
[[1988 AHSME Problems/Problem 12|Solution]]

==Problem 13==

If $\sin\ x\ =\ 3\ \cos\ x $ then what is $\sin\ x\ \cos\ x$?

$\textbf{(A)}\ \frac{1}{6}\qquad
\textbf{(B)}\ \frac{1}{5}\qquad
\textbf{(C)}\ \frac{2}{9}\qquad
\textbf{(D)}\ \frac{1}{4}\qquad
\textbf{(E)}\ \frac{3}{10} $   
  
[[1988 AHSME Problems/Problem 13|Solution]]

==Problem 14==

For any real number a and positive integer k, define

${a \choose k} = \frac{a(a-1)(a-2)\cdots(a-(k-1))}{k(k-1)(k-2)\cdots(2)(1)}$

What is

${-\frac{1}{2} \choose 100} \div {\frac{1}{2} \choose 100}$?

$\textbf{(A)}\ -199\qquad
\textbf{(B)}\ -197\qquad
\textbf{(C)}\ -1\qquad
\textbf{(D)}\ 197\qquad
\textbf{(E)}\ 199     $
  
[[1988 AHSME Problems/Problem 14|Solution]]

==Problem 15==

If $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1$, then $b$ is

$\textbf{(A)}\ -2\qquad
\textbf{(B)}\ -1\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ 1\qquad
\textbf{(E)}\ 2$     
  
[[1988 AHSME Problems/Problem 15|Solution]]

==Problem 16==

$ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, 
as in the figure. The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$. 
The ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ is

<asy>
defaultpen(linewidth(0.7)+fontsize(10));
pair H=origin, B=(1,-(1/sqrt(3))), C=(-1,-(1/sqrt(3))), A=(0,(2/sqrt(3))), E=(2,-(2/sqrt(3))), F=(-2,-(2/sqrt(3))), D=(0,(4/sqrt(3)));
draw(A--B--C--A^^D--E--F--D);
label("A'", A, N);
label("B'", B, SE);
label("C'", C, SW);
label("A", D, E);
label("B", E, E);
label("C", F, W);
<\asy>


$\textbf{(A)}\ \frac{1}{36}\qquad
\textbf{(B)}\ \frac{1}{6}\qquad
\textbf{(C)}\ \frac{1}{4}\qquad
\textbf{(D)}\ \frac{\sqrt{3}}{4}\qquad
\textbf{(E)}\ \frac{9+8\sqrt{3}}{36} $    
  
[[1988 AHSME Problems/Problem 16|Solution]]

==Problem 17==

If $|x| + x + y = 10$ and $x + |y| - y = 12$, find $x + y$

$\textbf{(A)}\ -2\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ \frac{18}{5}\qquad
\textbf{(D)}\ \frac{22}{3}\qquad
\textbf{(E)}\ 22     $
  
[[1988 AHSME Problems/Problem 17|Solution]]

==Problem 18==

At the end of a professional bowling tournament, the top 5 bowlers have a playoff. 
First #5 bowls #4. The loser receives $5$th prize and the winner bowls #3 in another game. 
The loser of this game receives $4$th prize and the winner bowls #2. 
The loser of this game receives $3$rd prize and the winner bowls #1. 
The winner of this game gets 1st prize and the loser gets 2nd prize. 
In how many orders can bowlers #1 through #5 receive the prizes?

\textbf{(A)}\ 10\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 120\qquad
\textbf{(E)}\ \text{none of these}     
  
[[1988 AHSME Problems/Problem 18|Solution]]

==Problem 19==

Simplify

$\frac{bx(a^2x^2 + 2a^2y^2 + b^2y^2) + ay(a^2x^2 + 2b^2x^2 + b^2y^2)}{bx + ay}$

$\textbf{(A)}\ a^2x^2 + b^2y^2\qquad
\textbf{(B)}\ (ax + by)^2\qquad
\textbf{(C)}\ (ax + by)(bx + ay)\qquad\\
\textbf{(D)}\ 2(a^2x^2+b^2y^2)\qquad
\textbf{(E)}\ (bx+ay)^2$   
  
[[1988 AHSME Problems/Problem 19|Solution]]

==Problem 20==

In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints
of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown 
in the second figure. The ratio of height to base, $XY / YZ$, in this rectangle is  

<asy>
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(0,1), B=(0,-1), C=(2,-1), D=(2,1), E=(1,-1), F=(1,1), G=(.8,.6);
pair X=(4,sqrt(5)), Y=(4,-sqrt(5)), Z=(4+2/sqrt(5),-sqrt(5)), W=(4+2/sqrt(5),sqrt(5)), T=(4,0), U=(4+2/sqrt(5),-4/sqrt(5)), V=(4+2/sqrt(5),1/sqrt(5));
draw(A--B--C--D--A^^B--F^^E--D^^A--G^^rightanglemark(A,G,F));
draw(X--Y--Z--W--X^^T--V--X^^Y--U);
label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, S);
label("F", F, N);
label("G", G, E);
label("X", X, NW);
label("Y", Y, SW);
label("Z", Z, SE);
label("W", W, NE);
 (Error making remote request. Unknown error_msg)

Solution

Problem 21

The complex number satisfies . What is ? Note: if , then .

Solution

Problem 22

For how many integers does a triangle with side lengths and have all its angles acute?

Solution

Problem 23

The six edges of a tetrahedron measure and units. If the length of edge is , then the length of edge is

Solution

Problem 24

An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is , and one of the base angles is . Find the area of the trapezoid.

Solution

Problem 25

and are pairwise disjoint sets of people. The average ages of people in the sets and are and respectively. Find the average age of the people in set .

Solution

Problem 26

Suppose that and are positive numbers for which

What is the value of ?

Solution

Problem 27

In the figure, , and is tangent to the circle with center and diameter . In which one of the following cases is the area of an integer?

Solution

Problem 28

An unfair coin has probability of coming up heads on a single toss. Let be the probability that, in independent toss of this coin, heads come up exactly times. If , then

Solution

Problem 29

You plot weight against height for three of your friends and obtain the points . If and , which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.

Solution

Problem 30

Let . Give , consider the sequence defined by for all . For how many real numbers will the sequence take on only a finite number of different values?

Solution