# Difference between revisions of "1974 AHSME Problems"

 1974 AHSME (Answer Key)Printable version: 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

If $x\not=0$ or $4$ and $y\not=0$ or $6$, then $\frac{2}{x}+\frac{3}{y}=\frac{1}{2}$ is equivalent to

$\mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } \frac{x}{2}+\frac{y}{3}=2 \qquad$

$\mathrm{(D) \ } \frac{4y}{y-6}=x \qquad \mathrm{(E) \ }\text{none of these}$

## Problem 2

Let $x_1$ and $x_2$ be such that $x_1\not=x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals

$\mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \ } \frac{b}{3} \qquad \mathrm{(D) \ } 2b \qquad \mathrm{(E) \ }-\frac{b}{3}$

## Problem 3

The coefficient of $x^7$ in the polynomial expansion of

$$(1+2x-x^2)^4$$

is

$\mathrm{(A)\ } -8 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 6 \qquad \mathrm{(D) \ } -12 \qquad \mathrm{(E) \ }\text{none of these}$

## Problem 4

What is the remainder when $x^{51}+51$ is divided by $x+1$?

$\mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 49 \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ }51$

## Problem 5

Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^\circ$ and $\measuredangle ADC=68^\circ$, find $\measuredangle EBC$.

$\mathrm{(A)\ } 66^\circ \qquad \mathrm{(B) \ }68^\circ \qquad \mathrm{(C) \ } 70^\circ \qquad \mathrm{(D) \ } 88^\circ \qquad \mathrm{(E) \ }92^\circ$

## Problem 6

For positive real numbers $x$ and $y$ define $x*y=\frac{x\cdot y}{x+y}$' then

$\mathrm{(A)\ } \text{*" is commutative but not associative} \qquad$

$\mathrm{(B) \ }\text{*" is associative but not commutative} \qquad$

$\mathrm{(C) \ } \text{*" is neither commutative nor associative} \qquad$

$\mathrm{(D) \ } \text{*" is commutative and associative} \qquad$

$\mathrm{(E) \ }\text{none of these} \qquad$

## Problem 7

A town's population increased by $1,200$ people, and then this new population decreased by $11\%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population?

$\mathrm{(A)\ } 1,200 \qquad \mathrm{(B) \ }11,200 \qquad \mathrm{(C) \ } 9,968 \qquad \mathrm{(D) \ } 10,000 \qquad \mathrm{(E) \ }\text{none of these}$

## Problem 8

What is the smallest prime number dividing the sum $3^{11}+5^{13}$?

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 5 \qquad \mathrm{(D) \ } 3^{11}+5^{13} \qquad \mathrm{(E) \ }\text{none of these}$

## Problem 9

The integers greater than one are arranged in five columns as follows:

$$\begin{tabular}{c c c c c}\ & 2 & 3 & 4 & 5\\ 9 & 8 & 7 & 6 &\ \\ \ & 10 & 11 & 12 & 13\\ 17 & 16 & 15 & 14 &\ \\ \ & . & . & . & .\\ \end{tabular}$$

(Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.)

In which column will the number $1,000$ fall?

$\mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \ } \text{third} \qquad \mathrm{(D) \ } \text{fourth} \qquad \mathrm{(E) \ }\text{fifth}$

## Problem 10

What is the smallest integral value of $k$ such that

$$2x(kx-4)-x^2+6=0$$

has no real roots?

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

## Problem 11

If $(a, b)$ and $(c, d)$ are two points on the line whose equation is $y=mx+k$, then the distance between $(a, b)$ and $(c, d)$, in terms of $a, c,$ and $m$ is

$\mathrm{(A)\ } |a-c|\sqrt{1+m^2} \qquad \mathrm{(B) \ }|a+c|\sqrt{1+m^2} \qquad \mathrm{(C) \ } \frac{|a-c|}{\sqrt{1+m^2}} \qquad$

$\mathrm{(D) \ } |a-c|(1+m^2) \qquad \mathrm{(E) \ }|a-c|\,|m|$

## Problem 12

If $g(x)=1-x^2$ and $f(g(x))=\frac{1-x^2}{x^2}$ when $x\not=0$, then $f(1/2)$ equals

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

## Problem 13

Which of the following is equivalent to "If P is true, then Q is false."?

$\mathrm{(A)\ } \text{P is true or Q is false."} \qquad$

$\mathrm{(B) \ }\text{If Q is false then P is true."} \qquad$

$\mathrm{(C) \ } \text{If P is false then Q is true."} \qquad$

$\mathrm{(D) \ } \text{If Q is true then P is false."} \qquad$

$\mathrm{(E) \ }\text{If Q is true then P is true."} \qquad$

## Problem 14

Which statement is correct?

$\mathrm{(A)\ } \text{If } x<0, \text{then } x^2>x. \qquad \mathrm{(B) \ } \text{If } x^2>0, \text{then } x>0.$

$\qquad \mathrm{(C) \ } \text{If } x^2>x, \text{then } x>0. \qquad \mathrm{(D) \ } \text{If } x^2>x, \text{then } x<0.$

$\qquad \mathrm{(E) \ }\text{If } x<1, \text{then } x^2

## Problem 15

If $x<-2$, then $|1-|1+x||$ equals

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

## Problem 16

A circle of radius $r$ is inscribed in a right isosceles triangle, and a circle of radius $R$ is circumscribed about the triangle. Then $R/r$ equals

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

## Problem 17

If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals

$\mathrm{(A)\ } -1024 \qquad \mathrm{(B) \ }-1024i \qquad \mathrm{(C) \ } 0 \qquad \mathrm{(D) \ } 1024 \qquad \mathrm{(E) \ }1024i$

## Problem 18

If $\log_8{3}=p$ and $\log_3{5}=q$, then, in terms of $p$ and $q$, $\log_{10}{5}$ equals

$\mathrm{(A)\ } pq \qquad \mathrm{(B) \ }\frac{3p+q}{5} \qquad \mathrm{(C) \ } \frac{1+3pq}{p+q} \qquad \mathrm{(D) \ } \frac{3pq}{1+3pq} \qquad \mathrm{(E) \ }p^2+q^2$

## Problem 19

In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is

$[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((.82,0)--(1,1)--(0,.76)--cycle); label("A", (0,0), S); label("B", (1,0), S); label("C", (1,1), N); label("D", (0,1), N); label("M", (0,.76), W); label("N", (.82,0), S);[/asy]$

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

## Problem 20

Let

$$T=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}.$$

Then

$\mathrm{(A)\ } T<1 \qquad \mathrm{(B) \ }T=1 \qquad \mathrm{(C) \ } 12 \qquad$

$\mathrm{(E) \ }T=\frac{1}{(3-\sqrt{8})(\sqrt{8}-\sqrt{7})(\sqrt{7}-\sqrt{6})(\sqrt{6}-\sqrt{5})(\sqrt{5}-2)}$

## Problem 21

In a geometric series of positive terms the difference between the fifth and fourth terms is $576$, and the difference between the second and first terms is $9$. What is the sum of the first five terms of this series?

$\mathrm{(A)\ } 1061 \qquad \mathrm{(B) \ }1023 \qquad \mathrm{(C) \ } 1024 \qquad \mathrm{(D) \ } 768 \qquad \mathrm{(E) \ }\text{none of these}$

## Problem 22

The minimum of $\sin\frac{A}{2}-\sqrt{3}\cos\frac{A}{2}$ is attained when $A$ is

$\mathrm{(A)\ } -180^\circ \qquad \mathrm{(B) \ }60^\circ \qquad \mathrm{(C) \ } 120^\circ \qquad \mathrm{(D) \ } 0^\circ \qquad \mathrm{(E) \ }\text{none of these}$

## Problem 23

In the adjoining figure $TP$ and $T'Q$ are parallel tangents to a circle of radius $r$, with $T$ and $T'$ the points of tangency. $PT''Q$ is a third tangent with $T'''$ as a point of tangency. If $TP=4$ and $T'Q=9$ then $r$ is

$[asy] unitsize(45); pair O = (0,0); pair T = dir(90); pair T1 = dir(270); pair T2 = dir(25); pair P = (.61,1); pair Q = (1.61, -1); draw(unitcircle); dot(O); label("O",O,W); label("T",T,N); label("T'",T1,S); label("T''",T2,NE); label("P",P,NE); label("Q",Q,S); draw(O--T2); label("r",midpoint(O--T2),NW); draw(T--P); label("4",midpoint(T--P),N); draw(T1--Q); label("9",midpoint(T1--Q),S); draw(P--Q);[/asy]$

$\mathrm{(A)\ } 25/6 \qquad \mathrm{(B) \ } 6 \qquad \mathrm{(C) \ } 25/4 \qquad$

$\mathrm{(D) \ } \text{a number other than }25/6, 6, 25/4 \qquad$

$\mathrm{(E) \ }\text{not determinable from the given information}$

## Problem 24

A fair die is rolled six times. The probability of rolling at least a five at least five times is

$\mathrm{(A)\ } \frac{13}{729} \qquad \mathrm{(B) \ }\frac{12}{729} \qquad \mathrm{(C) \ } \frac{2}{729} \qquad \mathrm{(D) \ } \frac{3}{729} \qquad \mathrm{(E) \ }\text{none of these}$

## Problem 25

In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to

$[asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label("A", (0,0), S); label("B", (5,0), S); label("C", (6,3), NE); label("D", (1,3), NW); label("P", (10,0), E); label("Q", (-5,0), W); label("M", (.5,1.5), NW); label("N", (5.65, 1.5), NE); label("O", (3.4,1.75));[/asy]$

$\mathrm{(A)\ } k \qquad \mathrm{(B) \ }\frac{6k}{5} \qquad \mathrm{(C) \ } \frac{9k}{8} \qquad \mathrm{(D) \ } \frac{5k}{4} \qquad \mathrm{(E) \ }2k$

## Problem 26

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is

$\mathrm{(A)\ } 100 \qquad \mathrm{(B) \ }125 \qquad \mathrm{(C) \ } 123 \qquad \mathrm{(D) \ } 30 \qquad \mathrm{(E) \ }\text{none of these}$

## Problem 27

If $f(x)=3x+2$ for all real $x$, then the statement: "$|f(x)+4| whenever $|x+2| and $a>0$ and $b>0$" is true when

$\mathrm{(A)}\ b\le a/3\qquad\mathrm{(B)}\ b > a/3\qquad\mathrm{(C)}\ a\le b/3\qquad\mathrm{(D)}\ a > b/3\\ \qquad\mathrm{(E)}\ \text{The statement is never true.}$

## Problem 28

Which of the following is satisfied by all numbers $x$ of the form

$$x=\frac{a_1}{3}+\frac{a_2}{3^2}+\cdots+\frac{a_{25}}{3^{25}}$$

where $a_1$ is $0$ or $2$, $a_2$ is $0$ or $2$,...,$a_{25}$ is $0$ or $2$?

$\mathrm{(A)\ } 0\le x<1/3 \qquad \mathrm{(B) \ } 1/3\le x<2/3 \qquad \mathrm{(C) \ } 2/3\le x<1 \qquad$

$\mathrm{(D) \ } 0\le x<1/3 \text{ or }2/3\le x<1 \qquad \mathrm{(E) \ }1/2\le x\le 3/4$

## Problem 29

For $p=1, 2, \cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\cdots+S_{10}$ is

$\mathrm{(A)\ } 80000 \qquad \mathrm{(B) \ }80200 \qquad \mathrm{(C) \ } 80400 \qquad \mathrm{(D) \ } 80600 \qquad \mathrm{(E) \ }80800$

## Problem 30

A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $R$ is the ratio of the lesser part to the greater part, then the value of

$$R^{[R^{(R^2+R^{-1})}+R^{-1}]}+R^{-1}$$

is

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