1984 AHSME Problems

Problem 1

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

$\mathrm{(A) \ }62,500 \qquad \mathrm{(B) \ }1,000 \qquad \mathrm{(C) \ } 500\qquad \mathrm{(D) \ }250 \qquad \mathrm{(E) \ } \frac{1}{2}$

Solution

Problem 2

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

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

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

Solution

Problem 3

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

$\mathrm{(A) \ }100<n\leq110 \qquad \mathrm{(B) \ }110<n\leq120 \qquad \mathrm{(C) \ } 120<n\leq130 \qquad \mathrm{(D) \ }130<n\leq140 \qquad \mathrm{(E) \ } 140<n\leq150$

Solution

Problem 4

Points $B, C, F, E$ are picked on a circle such that $BC||EF$ . When $BC$ is extended to the left, point $A$ is marked outside the circle such that $AB=4$ and $BC=5$ . When $EF$ is extended to the left, point $D$ is marked outside the circle such that $DE=3$ . $AD$ is perpendicular to both $AC$ and $DF$ . Find the length of $EF$ .

A = HOOT! B = BOOT! C = FROOT! D = [I am] GROOT! E = NOOT!

Solution

Problem 5

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

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

Solution

Problem 6

In a certain school, there are $3$ times as many boys as girls and $9$ 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

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

Solution

Problem 7

When Dave walks to school, he averages $90$ steps per minute, and each of his steps is $75$ 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?

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

Solution

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

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

Solution

Problem 9

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

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

Solution

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

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

Solution

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 a number $x\not=0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then $y$ equals

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

Solution

Problem 12

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

$a_1=2$

$a_{n+1}=a_n+2n$

where $n\geq1$ .

Then $a_{100}$ equals

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

Solution

Problem 13

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

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

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

Solution

Problem 14

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

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

Solution

Problem 15

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

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

Solution

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

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

Solution

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:

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

Solution

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

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

Solution

Problem 19

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

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

Solution

Problem 20

The number of the distinct solutions to the equation

$|x-|2x+1||=3$ is

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

Solution

Problem 21

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

$ab+bc=44$

$ac+bc=23$

is

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

Solution

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 numbers of $t$ is graphed on the plane, the graph is

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

Solution

Problem 23

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

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

Solution

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

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

Solution

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

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

Solution

Problem 26

In the obtuse triangle $ABC$ with $\angle C>90^\circ$ , $AM=MB$ , $MD\perp BC$ , and $EC\perp BC$ ( $D$ is on $BC$ , $E$ is on $AB$ , and $M$ is on $EB$ ). If the area of $\triangle ABC$ is $24$ , then the area of $\triangle BED$ is

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

Solution

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 $AC$ .

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

Solution

Problem 28

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

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

Solution

Problem 29

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

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

Solution

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

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

Solution

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1984 AHSME Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30