# 1991 AHSME Problems

## Problem 1

If for any three distinct numbers $a$, $b$, and $c$ we define $f(a,b,c)=\frac{c+a}{c-b}$, then $f(1,-2,-3)$ is

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

## Problem 2

$|3-\pi|=$

$\textbf{(A)\ }\frac{1}{7}\qquad\textbf{(B)\ }0.14\qquad\textbf{(C)\ }3-\pi\qquad\textbf{(D)\ }3+\pi\qquad\textbf{(E)\ }\pi-3$

## Problem 3

$(4^{-1}-3^{-1})^{-1}=$

(A) $-12$ (B) $-1$ (C) $\frac{1}{12}$ (D) $1$ (E) $12$

## Problem 4

Which of the following triangles cannot exist?

(A) An acute isosceles triangle (B) An isosceles right triangle (C) An obtuse right triangle (D) A scalene right triangle (E) A scalene obtuse tr iangle

## Problem 5

$[asy] draw((0,0)--(2,2)--(2,1)--(5,1)--(5,-1)--(2,-1)--(2,-2)--cycle,dot); MP("A",(0,0),W);MP("B",(2,2),N);MP("C",(2,1),S);MP("D",(5,1),NE);MP("E",(5,-1),SE);MP("F",(2,-1),NW);MP("G",(2,-2),S); MP("5",(2,1.5),E);MP("5",(2,-1.5),E);MP("20",(3.5,1),N);MP("20",(3.5,-1),S);MP("10",(5,0),E); [/asy]$

In the arrow-shaped polygon [see figure], the angles at vertices $A,C,D,E$ and $F$ are right angles, $BC=FG=5, CD=FE=20, DE=10$, and $AB=AG$. The area of the polygon is closest to $\text{(A) } 288\quad \text{(B) } 291\quad \text{(C) } 294\quad \text{(D) } 297\quad \text{(E) } 300$

## Problem 6

If $x\geq 0$, then $\sqrt{x\sqrt{x\sqrt{x}}}=$

(A) $x\sqrt{x}$ (B) $x \sqrt[4]{x}$ (C) $\sqrt[8]{x}$ (D) $\sqrt[8]{x^3}$ (E) $\sqrt[8]{x^7}$

## Problem 7

If $x=\frac{a}{b}$, $a\neq b$ and $b\neq 0$, then $\frac{a+b}{a-b}=$

(A) $\frac{x}{x+1}$ (B) $\frac{x+1}{x-1}$ (C) $1$ (D) $x-\frac{1}{x}$ (E) $x+\frac{1}{x}$

## Problem 8

Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?

(A) $\frac{\sqrt{216}}{\pi}$ (B) $\sqrt{\frac{216}{\pi}}$ (C) $\sqrt{\frac{2160}{\pi}}$ (D) $\frac{216}{\pi}$ (E) $\frac{2160}{\pi}$

## Problem 9

From time $t=0$ to time $t=1$ a population increased by $i\%$, and from time $t=1$ to time $t=2$ the population increased by $j\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by

$\text{(A) (i+j)\%} \quad \text{(B) } ij\%\quad \text{(C) } (i+ij)\%\quad \text{(D) } \left(i+j+\frac{ij}{100}\right)\%\quad \text{(E) } \left(i+j+\frac{i+j}{100}\right)\%$

## Problem 10

Point $P$ is $9$ units from the center of a circle of radius $15$. How many different chords of the circle contain $P$ and have integer lengths?

(A) 11 (B) 12 (C) 13 (D) 14 (E) 29

## Problem 11

Jack and Jill run 10 km. They start at the same point, run 5 km up a hill, and reurn to the starting point by the same route. Jack has a 10 minute head start and runs at the rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill. How far from the top of the hill are they when they pass each other going in opposite directions (in km)?

$\text{(A) } \frac{5}{4}\quad \text{(B) } \frac{35}{27}\quad \text{(C) } \frac{27}{20}\quad \text{(D) } \frac{7}{3}\quad \text{(E) } \frac{28}{49}$

## Problem 12

The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of integers. Let $m$ be the measure of the largest interior angle of the hexagon. The largest possible value of $m$, in degrees, is

(A) 165 (B) 167 (C) 170 (D) 175 (E) 179

## Problem 13

Horses $X,Y$ and $Z$ are entered in a three-horse race in which ties are not possible. The odds against $X$ winning are $3:1$ and the odds against $Y$ winning are $2:3$, what are the odds against $Z$ winning? (By "odds against $H$ winning are $p:q$" we mean the probability of $H$ winning the race is $\frac{q}{p+q}$.)

$\text{(A) } 3:20\quad \text{(B) } 5:6\quad \text{(C) } 8:5\quad \text{(D) } 17:3\quad \text{(E) } 20:3$

## Problem 14

If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be

(A) $200$ (B) $201$ (C) $202$ (D) $203$ (E) $204$

## Problem 15

A circular table has 60 chairs around it. There are $N$ people seated at this table in such a way that the next person seated must sit next to someone. What is the smallest possible value for $N$?

$\text{(A) } 15\quad \text{(B) } 20\quad \text{(C) } 30\quad \text{(D) } 40\quad \text{(E) } 58$

## Problem 16

One hundred students at Century High School participated in the AHSME last year, and their mean score was 100. The number of non-seniors taking the AHSME was $50\%$ more than the number of seniors, and the mean score of the seniors was $50\%$ higher than that of the non-seniors. What was the mean score of the seniors?

(A) $100$ (B) $112.5$ (C) $120$ (D) $125$ (E) $150$

## Problem 17

A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following 2 properties:

(a) It is a palindrome (b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome.

How many years in the millenium between 1000 and 2000 have properties (a) and (b)?

$\text{(A) } 1\quad \text{(B) } 2\quad \text{(C) } 3\quad \text{(D) } 4\quad \text{(E) } 5$

## Problem 18

If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a

(A) right triangle (B) circle (C) hyperbola (D) line (E) parabola

## Problem 19

$[asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP("C",(0,0),SW);MP("A",(0,3),NW);MP("B",(4,0),S);MP("E",(7,0),SE);MP("D",(7,10),NE); [/asy]$

Triangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$. Triangle $ABD$ has a right angle at $A$ and $AD=12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $$\frac{DE}{DB}=\frac{m}{n},$$ where $m$ and $n$ are relatively prime positive integers, then $m+n=$

$\text{(A) } 25\quad \text{(B) } 128\quad \text{(C) } 153\quad \text{(D) } 243\quad \text{(E) } 256$

## Problem 20

The sum of all real $x$ such that $(2^x-4)^3+(4^x-2)^3=(4^x+2^x-6)^3$ is

(A) 3/2 (B) 2 (C) 5/2 (D) 3 (E) 7/2

## Problem 21

For all real numbers $x$ except $x=0$ and $x=1$ the function $f(x)$ is defined by $f(x/(1-x))=1/x$. Suppose $0\leq t\leq \pi/2$. What is the value of $f(\sec^2t)$?

$\text{(A) } \sin^2\theta\quad \text{(B) } \cos^2\theta\quad \text{(C) } \tan^2\theta\quad \text{(D) } \cot^2\theta\quad \text{(E) } \csc^2\theta$

## Problem 22

$[asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP("B",(-8/3,16*sqrt(2)/3),W);MP("B'",(8/3,16*sqrt(2)/3),E); MP("A",(-4/3,8*sqrt(2)/3),W);MP("A'",(4/3,8*sqrt(2)/3),E); MP("P",(0,0),S); [/asy]$

Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle $B$ and $B'$ on the larger circle. If $PA=AB=4$, then the area of the smaller circle is

$\text{(A) } 1.44\pi\quad \text{(B) } 2\pi\quad \text{(C) } 2.56\pi\quad \text{(D) } \sqrt{8}\pi\quad \text{(E) } 4\pi$

## Problem 23

$[asy] draw((0,0)--(0,2)--(2,2)--(2,0)--cycle,dot); draw((2,2)--(0,0)--(0,1)--cycle,dot); draw((0,2)--(1,0),dot); MP("B",(0,0),SW);MP("A",(0,2),NW);MP("D",(2,2),NE);MP("C",(2,0),SE); MP("E",(0,1),W);MP("F",(1,0),S);MP("H",(2/3,2/3),E);MP("I",(2/5,6/5),N); dot((1,0));dot((0,1));dot((2/3,2/3));dot((2/5,6/5)); [/asy]$

If $ABCD$ is a $2X2$ square, $E$ is the midpoint of $\overline{AB}$,$F$ is the midpoint of $\overline{BC}$,$\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is

$\text{(A) } \frac{1}{3}\quad \text{(B) } \frac{2}{5}\quad \text{(C) } \frac{7}{15}\quad \text{(D) } \frac{8}{15}\quad \text{(E) } \frac{3}{5}$

## Problem 24

The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?

(A) $y=\log_{10}\left(\frac{x+90}{9}\right)$ (B) $y=\log_{x}10$ (C) $y=\frac{1}{x+1}$ (D) $y=10^{-x}$ (E) $y=10^x$

## Problem 25

If $T_n=1+2+3+\cdots +n$ and $$P_n=\frac{T_2}{T_2-1}\cdot\frac{T_3}{T_3-1}\cdot\frac{T_4}{T_4-1}\cdot\cdots\cdot\frac{T_n}{T_n-1}$$ for $n=2,3,4,\cdots,$ then $P_{1991}$ is closest to which of the following numbers?

$\text{(A) } 2.0\quad \text{(B) } 2.3\quad \text{(C) } 2.6\quad \text{(D) } 2.9\quad \text{(E) } 3.2$

## Problem 26

An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\{1,2,...,n\}$ and its first $k$ digits form an integer that is divisible by $k$ , for $k = 1,2,...,n$. For example, $321$ is a cute $3$-digit integer because $1$ divides $3$, $2$ divides $32$, and $3$ divides $321$. Howmany cute $6$-digit integers are there?

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

## Problem 27

If $x+\sqrt{x^2-1}+\frac{1}{x-\sqrt{x^2-1}}=20$ then $x^2+\sqrt{x^4-1}+\frac{1}{x^2+\sqrt{x^4-1}}=$

(A) $5.05$ (B) $20$ (C) $51.005$ (D) $61.25$ (E) $400$

## Problem 28

Initially an urn contains 100 white and 100 black marbles. Repeatedly 3 marbles are removed (at random) from the urn and replaced with some marbles from a pile outside the urn as follows: 3 blacks are replaced with 1 black, or 2 blacks and 1 white are replaced with a white and a black, or 1 black and 2 whites are replaced with 2 whites, or 3 whites are replaced with a black and a white. Which of the following could be the contents of the urn after repeated applications of this procedure?

(A) 2 black (B) 2 white (C) 1 black (D) 1 black and 1 white (E) 1 white

## Problem 29

Equilateral triangle $ABC$ has $P$ on $AB$ and $Q$ on $AC$. The triangle is folded along $PQ$ so that vertex $A$ now rests at $A'$ on side $BC$. If $BA'=1$ and $A'C=2$ then the length of the crease $PQ$ is

(A) $\frac{8}{5}$ (B) $\frac{7}{20}\sqrt{21}$ (C) $\frac{1+\sqrt{5}}{2}$ (D) $\frac{13}{8}$ (E) $\sqrt{3}$

## Problem 30

For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\cup B\cup C)$ and $|A|=|B|=100$, then what is the minimum possible value of $|A\cap B\cap C|$?

(A) 96 (B) 97 (C) 98 (D) 99 (E) 100