1988 AHSME Problems

Problem 1

$\sqrt{8}+\sqrt{18}=$

$\text{(A)}\ \sqrt{20} \qquad \text{(B)}\ 2(\sqrt{2}+\sqrt{3}) \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 5\sqrt{2}\qquad \text{(E)}\ 2\sqrt{13}$

Solution

Problem 2

Triangles $ABC$ and $XYZ$ are similar, with $A$ corresponding to $X$ and $B$ to $Y$ . If $AB=3, BC=4$ , and $XY=5$ , then $YZ$ is:

$\text{(A)}\ 3\frac{3}{4} \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 6\frac{1}{4} \qquad \text{(D)}\ 6\frac{2}{3} \qquad \text{(E)}\ 8$

Solution

Problem 3

Four rectangular paper strips of length $10$ and width $1$ 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)

$\textbf{(A)}\ 4\qquad \textbf{(B)}\ 1+2\sqrt{3}\qquad \textbf{(C)}\ 2\sqrt{5}\qquad \textbf{(D)}\ \frac{8+4\sqrt{3}}{3}\qquad \textbf{(E)}\ 5$

Solution

Problem 21

The complex number $z$ satisfies $z + |z| = 2 + 8i$ . What is $|z|^{2}$ ? Note: if $z = a + bi$ , then $|z| = \sqrt{a^{2} + b^{2}}$ .

$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$

Solution

Problem 22

For how many integers $x$ does a triangle with side lengths $10, 24$ and $x$ have all its angles acute?

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

Solution

Problem 23

The six edges of a tetrahedron $ABCD$ measure $7, 13, 18, 27, 36$ and $41$ units. If the length of edge $AB$ is $41$ , then the length of edge $CD$ is

$\textbf{(A)}\ 7\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36$

Solution

Problem 24

An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$ , and one of the base angles is $\arcsin(.8)$ . Find the area of the trapezoid.

$\textbf{(A)}\ 72\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 80\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ \text{not uniquely determined}$

Solution

Problem 25

$X, Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets $X, Y, Z, X \cup Y, X \cup Z$ and $Y \cup Z$ are $37, 23, 41, 29, 39.5$ and $33$ respectively. Find the average age of the people in set $X \cup Y \cup Z$ .

$\textbf{(A)}\ 33\qquad \textbf{(B)}\ 33.5\qquad \textbf{(C)}\ 33.6\overline{6}\qquad \textbf{(D)}\ 33.83\overline{3}\qquad \textbf{(E)}\ 34$

Solution

Problem 26

Suppose that $p$ and $q$ are positive numbers for which

$\log_{9}(p) = \log_{12}(q) = \log_{16}(p+q)$

What is the value of $\frac{q}{p}$ ?

$\textbf{(A)}\ \frac{4}{3}\qquad \textbf{(B)}\ \frac{1+\sqrt{3}}{2}\qquad \textbf{(C)}\ \frac{8}{5}\qquad \textbf{(D)}\ \frac{1+\sqrt{5}}{2}\qquad \textbf{(E)}\ \frac{16}{9}$

Solution

Problem 27

In the figure, $AB \perp BC, BC \perp CD$ , and $BC$ is tangent to the circle with center $O$ and diameter $AD$ . In which one of the following cases is the area of $ABCD$ an integer?

$[asy] defaultpen(fontsize(10pt)+linewidth(.8pt)); pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2)); draw(unitcircle); dot(O); draw(A--B--C--D--A); label("$A$",A,dir(A)); label("$B$",B,dir(B)); label("$C$",C,dir(C)); label("$D$",D,dir(D)); label("$O$",O,N); [/asy]$

$\textbf{(A)}\ AB=3, CD=1\qquad \textbf{(B)}\ AB=5, CD=2\qquad \textbf{(C)}\ AB=7, CD=3\qquad \textbf{(D)}\ AB=9, CD=4\qquad \textbf{(E)}\ AB=11, CD=5$

Solution

Problem 28

An unfair coin has probability $p$ of coming up heads on a single toss. Let $w$ be the probability that, in $5$ independent toss of this coin, heads come up exactly $3$ times. If $w = 144 / 625$ , then

$\textbf{(A)}\ p\text{ must be }{2/5}\\qquad \textbf{(B)}\ p\text{ must be }{3/5}\\qquad \textbf{(C)}\ p\text{ must be greater than }{3/5}\\qquad \textbf{(D)}\ p\text{ is not uniquely determined}\\qquad \textbf{(E)}\ \text{there is no value of } p \text{ for which }w ={ 144/625}$

Solution

Problem 29

You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$ . If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$ , 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.

$\textbf{(A)}\ \frac{y_{3}-y_{1}}{x_{3}-x_{1}}\qquad \textbf{(B)}\ \frac{(y_{2}-y_{1})-(y_{3}-y_{2})}{x_{3}-x_{1}}\qquad\\ \textbf{(C)}\ \frac{2y_{3}-y_{1}-y_{2}}{2x_{3}-x_{1}-x_{2}}\qquad \textbf{(D)}\ \frac{y_{2}-y_{1}}{x_{2}-x_{1}}+\frac{y_{3}-y_{2}}{x_{3}-x_{2}}\qquad\\ \textbf{(E)}\ \text{none of these}$

Solution

Problem 30

Let $f(x) = 4x - x^{2}$ . Give $x_{0}$ , consider the sequence defined by $x_{n} = f(x_{n-1})$ for all $n \ge 1$ . For how many real numbers $x_{0}$ will the sequence $x_{0}, x_{1}, x_{2}, \ldots$ take on only a finite number of different values?

$\textbf{(A)}\ \text{0}\qquad \textbf{(B)}\ \text{1 or 2}\qquad \textbf{(C)}\ \text{3, 4, 5 or 6}\qquad\\ \textbf{(D)}\ \text{more than 6 but finitely many}\qquad\\ \textbf{(E)}\ \text{infinitely many}$

Solution

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1988 AHSME Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30