# 1984 AHSME Problems

 1984 AHSME (Answer Key)Printable versions: Wiki • AoPS Resources • PDF Instructions 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. You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. Figures are not necessarily drawn to scale. 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

$\frac{1000^2}{252^2-248^2}$ equals

$\mathrm{(A) \ }62500 \qquad \mathrm{(B) \ }1000 \qquad \mathrm{(C) \ } 500\qquad \mathrm{(D) \ }250 \qquad \mathrm{(E) \ } \frac{1}{2}$

## Problem 2

If $x, y$, and $y-\frac{1}{x}$ are not $0$, then

$$\frac{x-\frac{1}{y}}{y-\frac{1}{x}}$$ equals

$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }\frac{x}{y} \qquad \mathrm{(C) \ } \frac{y}{x}\qquad \mathrm{(D) \ }\frac{x}{y}-\frac{y}{x} \qquad \mathrm{(E) \ } xy-\frac{1}{xy}$

## Problem 3

Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$. Then

$\mathrm{(A) \ }100

## Problem 4

A rectangle intersects a circle as shown: $AB = 4$, $BC = 5$ and $DE = 3$. Then $EF$ equals

$\mathrm{(A) \ }6 \qquad \mathrm{(B) \ }7 \qquad \mathrm{(C) \ }\frac{20}{3} \qquad \mathrm{(D) \ }8 \qquad \mathrm{(E) \ }9$

## Problem 5

The largest integer $n$ for which $n^{200}<5^{300}$ is

$\mathrm{(A) \ }8 \qquad \mathrm{(B) \ }9 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ }11 \qquad \mathrm{(E) \ } 12$

## Problem 6

In a certain school, there are three times as many boys as girls and nine times as many girls as teachers. Using the letters $b, g, t$ to represent the number of boys, girls and teachers, respectively, then the total number of boys, girls and teachers can be represented by the expression

$\mathrm{(A) \ }31b \qquad \mathrm{(B) \ }\frac{37}{27}b \qquad \mathrm{(C) \ } 13g \qquad \mathrm{(D) \ }\frac{37}{27}g \qquad \mathrm{(E) \ } \frac{37}{27}t$

## Problem 7

When Dave walks to school, he averages $90$ steps per minute, each of his steps $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. How long does it take Jack to get to school?

$\mathrm{(A) \ }14 \frac{2}{9} \text{ minutes} \qquad \mathrm{(B) \ }15 \text{ minutes}\qquad \mathrm{(C) \ } 18 \text{ minutes}\qquad \mathrm{(D) \ }20 \text{ minutes} \qquad \mathrm{(E) \ } 22 \frac{2}{9} \text{ minutes}$

## Problem 8

Figure $ABCD$ is a trapezoid with $AB||DC$, $AB=5$, $BC=3\sqrt{2}$, $\angle BCD=45^\circ$ and $\angle CDA=60^\circ$. The length of $DC$ is

$\mathrm{(A) \ }7+\frac{2}{3}\sqrt{3} \qquad \mathrm{(B) \ }8 \qquad \mathrm{(C) \ } 9 \frac{1}{2} \qquad \mathrm{(D) \ }8+\sqrt{3} \qquad \mathrm{(E) \ } 8+3\sqrt{3}$

## Problem 9

The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is

$\mathrm{(A) \ }31 \qquad \mathrm{(B) \ }30 \qquad \mathrm{(C) \ } 29 \qquad \mathrm{(D) \ }28 \qquad \mathrm{(E) \ } 27$

## Problem 10

Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i, {-2}+i$ and ${-1}-2i$. The fourth number is

$\mathrm{(A) \ }2+i \qquad \mathrm{(B) \ }2-i \qquad \mathrm{(C) \ } 1-2i \qquad \mathrm{(D) \ }{-1}+2i \qquad \mathrm{(E) \ } {-2}-i$

## Problem 11

A calculator has a key that replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let $y$ be the final result when one starts with an entry $x\not=0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then $y$ equals

$\mathrm{(A) \ }x^{((-2)^n)} \qquad \mathrm{(B) \ }x^{2n} \qquad \mathrm{(C) \ } x^{-2n} \qquad \mathrm{(D) \ }x^{-(2^n)} \qquad \mathrm{(E) \ } x^{((-1)^n2n)}$

## Problem 12

If the sequence $\{a_n\}$ is defined by

$$a_1=2,$$ $$a_{n+1}=a_n+2n \qquad (n \geq 1),$$

then $a_{100}$ equals

$\mathrm{(A) \ }9900 \qquad \mathrm{(B) \ }9902 \qquad \mathrm{(C) \ } 9904 \qquad \mathrm{(D) \ }10100 \qquad \mathrm{(E) \ } 10102$

## Problem 13

$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$ equals

$\mathrm{(A) \ }\sqrt{2}+\sqrt{3}-\sqrt{5} \qquad \mathrm{(B) \ }4-\sqrt{2}-\sqrt{3} \qquad \mathrm{(C) \ } \sqrt{2}+\sqrt{3}+\sqrt{6}-5 \qquad$

$\mathrm{(D) \ }\frac{1}{2}(\sqrt{2}+\sqrt{5}-\sqrt{3}) \qquad \mathrm{(E) \ } \frac{1}{3}(\sqrt{3}+\sqrt{5}-\sqrt{2})$

## Problem 14

The product of all real roots of the equation $x^{\log_{10}{x}}=10$ is

$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }-1 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ }10^{-1} \qquad \mathrm{(E) \ } \text{None of these}$

## Problem 15

If $\sin{2x}\sin{3x}=\cos{2x}\cos{3x}$, then one value for $x$ is

$\mathrm{(A) \ }18^\circ \qquad \mathrm{(B) \ }30^\circ \qquad \mathrm{(C) \ } 36^\circ \qquad \mathrm{(D) \ }45^\circ \qquad \mathrm{(E) \ } 60^\circ$

## Problem 16

The function $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x$. If the equation $f(x)=0$ has exactly four distinct real roots, then the sum of these roots is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ } 8$

## Problem 17

A right triangle $ABC$ with hypotenuse $AB$ has side $AC=15$. Altitude $CH$ divides $AB$ into segments $AH$ and $HB$, with $HB=16$. The area of $\triangle ABC$ is

$\mathrm{(A) \ }120 \qquad \mathrm{(B) \ }144 \qquad \mathrm{(C) \ } 150 \qquad \mathrm{(D) \ }216 \qquad \mathrm{(E) \ } 144\sqrt{5}$

## Problem 18

A point $(x, y)$ is to be chosen in the coordinate plane so that it is equally distant from the $x$-axis, the $y$-axis, and the line $x+y=2$. Then $x$ is

$\mathrm{(A) \ }\sqrt{2}-1 \qquad \mathrm{(B) \ }\frac{1}{2} \qquad \mathrm{(C) \ } 2-\sqrt{2} \qquad \mathrm{(D) \ }1 \qquad \mathrm{(E) \ } \text{not uniquely determined}$

## Problem 19

A box contains $11$ balls, numbered $1, 2, 3,\ldots,11$. If $6$ balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd?

$\mathrm{(A) \ }\frac{100}{231} \qquad \mathrm{(B) \ }\frac{115}{231} \qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ }\frac{118}{231} \qquad \mathrm{(E) \ } \frac{6}{11}$

## Problem 20

The number of the distinct solutions of the equation $|{x-|2x+1|}|=3$ is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ } 4$

## Problem 21

The number of triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations

$$ab+bc=44,$$ $$ac+bc=23,$$

is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ } 4$

## Problem 22

Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y=ax^2+bx+c$. If the set of the vertices $(x_t, y_t)$ for all real values of $t$ is graphed on the plane, the graph is

$\mathrm{(A) \ } \text{a straight line} \qquad \mathrm{(B) \ } \text{a parabola} \qquad \mathrm{(C) \ } \text{part, but not all, of a parabola} \qquad \mathrm{(D) \ } \text{one branch of a hyperbola} \qquad \mathrm{(E) \ } \text{none of these}$

## Problem 23

$\frac{\sin{10^\circ}+\sin{20^\circ}}{\cos{10^\circ}+\cos{20^\circ}}$ equals

$\mathrm{(A) \ }\tan{10^\circ}+\tan{20^\circ} \qquad \mathrm{(B) \ }\tan{30^\circ} \qquad \mathrm{(C) \ } \frac{1}{2}(\tan{10^\circ}+\tan{20^\circ}) \qquad \mathrm{(D) \ }\tan{15^\circ} \qquad \mathrm{(E) \ } \frac{1}{4}\tan{60^\circ}$

## Problem 24

If $a$ and $b$ are positive real numbers and each of the equations $x^2+ax+2b=0$ and $x^2+2bx+a=0$ has real roots, then the smallest possible value of $a+b$ is

$\mathrm{(A) \ }2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ } 6$

## Problem 25

The total area of all the faces of a rectangular solid is $22 \ \text{cm}^2$, and the total length of all its edges is $24 \ \text{cm}$. Then the length in cm of any one of its interior diagonals is

$\mathrm{(A) \ }\sqrt{11} \qquad \mathrm{(B) \ }\sqrt{12} \qquad \mathrm{(C) \ } \sqrt{13}\qquad \mathrm{(D) \ }\sqrt{14} \qquad \mathrm{(E) \ } \text{not uniquely determined}$

## Problem 26

In the obtuse triangle $ABC$, $AM=MB$, $MD\perp BC$, $EC\perp BC$. If the area of $\triangle ABC$ is $24$, then the area of $\triangle BED$ is

$\mathrm{(A) \ }9 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 15 \qquad \mathrm{(D) \ }18 \qquad \mathrm{(E) \ } \text{not uniquely determined}$

## Problem 27

In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB\perp AC$, $AF\perp BC$, and $BD=DC=FC=1$. Find $AC$.

$\mathrm{(A) \ }\sqrt{2} \qquad \mathrm{(B) \ }\sqrt{3} \qquad \mathrm{(C) \ } \sqrt[3]{2} \qquad \mathrm{(D) \ }\sqrt[3]{3} \qquad \mathrm{(E) \ } \sqrt[4]{3}$

## Problem 28

The number of distinct pairs of integers $(x, y)$ such that $0 and $\sqrt{1984}=\sqrt{x}+\sqrt{y}$ is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ }4 \qquad \mathrm{(E) \ } 7$

## Problem 29

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x, y)$ which satisfy $(x-3)^2+(y-3)^2=6$.

$\mathrm{(A) \ }3+2\sqrt{2} \qquad \mathrm{(B) \ }2+\sqrt{3} \qquad \mathrm{(C) \ } 3\sqrt{3} \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ } 6+2\sqrt{3}$

## Problem 30

For any complex number $w=a+bi$, $|w|$ is defined to be the real number $\sqrt{a^2+b^2}$. If $w=\cos40^\circ+i\sin40^\circ$, then $|w+2w^2+3w^3+...+9w^9|^{-1}$ equals

$\mathrm{(A) \ }\frac{1}{9}\sin40^\circ \qquad \mathrm{(B) \ }\frac{2}{9}\sin20^\circ \qquad \mathrm{(C) \ } \frac{1}{9}\cos40^\circ \qquad \mathrm{(D) \ }\frac{1}{18}\cos20^\circ \qquad \mathrm{(E) \ } \text{none of these}$