1990 AHSME Problems

Problem 1

If $\dfrac{\frac{x}{4}}{2}=\dfrac{4}{\frac{x}{2}}$ , then $x=$

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

Solution

Problem 2

$\left(\frac{1}{4}\right)^{-\tfrac{1}{4}}=$

$\text{(A) } -16\quad \text{(B) } -\sqrt{2}\quad \text{(C) } -\frac{1}{16}\quad \text{(D) } \frac{1}{256}\quad \text{(E) } \sqrt{2}$

Solution

Problem 3

The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is $75^\circ$ , then the largest angle is

$\text{(A) } 95^\circ\quad \text{(B) } 100^\circ\quad \text{(C) } 105^\circ\quad \text{(D) } 110^\circ\quad \text{(E) } 115^\circ$

Solution

Problem 4

$[asy] draw((0,0)--(16,0)--(21,5*sqrt(3))--(5,5*sqrt(3))--cycle,dot); draw((5,5*sqrt(3))--(1,5*sqrt(3))--(16,0),dot); MP("A",(0,0),S);MP("B",(16,0),S);MP("C",(21,5sqrt(3)),NE);MP("D",(5,5sqrt(3)),N);MP("E",(1,5sqrt(3)),N); MP("16",(8,0),S);MP("10",(18.5,5sqrt(3)/2),E);MP("4",(3,5sqrt(3)),N); dot((4,4sqrt(3))); MP("F",(4,4sqrt(3)),W); [/asy]$

Let $ABCD$ be a parallelogram with $\angle{ABC}=120^\circ, AB=6$ and $BC=10.$ Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4.$ If $\overline{BE}$ intersects $\overline{AD}$ at $F$ , then $FD$ is closest to

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

Solution

Problem 5

Which of these numbers is largest?

$\text{(A) } \sqrt{\sqrt[3]{5\cdot 6}}\quad \text{(B) } \sqrt{6\sqrt[3]{5}}\quad \text{(C) } \sqrt{5\sqrt[3]{6}}\quad \text{(D) } \sqrt[3]{5\sqrt{6}}\quad \text{(E) } \sqrt[3]{6\sqrt{5}}$

Solution

Problem 6

Points $A$ and $B$ are $5$ units apart. How many lines in a given plane containing $A$ and $B$ are $2$ units from $A$ and $3$ units from $B$ ?

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

Solution

Problem 7

A triangle with integral sides has perimeter $8$ . The area of the triangle is

$\text{(A) } 2\sqrt{2}\quad \text{(B) } \frac{16}{9}\sqrt{3}\quad \text{(C) }2\sqrt{3} \quad \text{(D) } 4\quad \text{(E) } 4\sqrt{2}$

Solution

Problem 8

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

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

Solution

Problem 9

Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is

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

Solution

Problem 10

An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?

$\text{(A) } 328\quad \text{(B) } 329\quad \text{(C) } 330\quad \text{(D) } 331\quad \text{(E) } 332$

Solution

Problem 11

How many positive integers less than $50$ have an odd number of positive integer divisors?

$\text{(A) } 3\quad \text{(B) } 5\quad \text{(C) } 7\quad \text{(D) } 9\quad \text{(E) } 11$

Solution

Problem 12

Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$ . If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$

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

Solution

Problem 13

If the following instructions are carried out by a computer, which value of $X$ will be printed because of instruction $5$ ?

1. START  AT  AND  AT .  
2. INCREASE THE VALUE OF  BY .   
3. INCREASE THE VALUE OF  BY THE VALUE OF . 
4. IF  IS AT LEAST ,   
       THEN GO TO INSTRUCTION ;  
       OTHERWISE, GO TO INSTRUCTION .  
       AND PROCEED FROM THERE.  
5. PRINT THE VALUE OF .  
6.STOP.

$\text{(A) } 19\quad \text{(B) } 21\quad \text{(C) } 23\quad \text{(D) } 199\quad \text{(E) } 201$

Solution

Problem 14

$[asy] draw(circle((0,0),1),black); draw((0,1)--(cos(pi/14),-sin(pi/14))--(-cos(pi/14),-sin(pi/14))--cycle,dot); draw((-cos(pi/14),-sin(pi/14))--(0,-1/cos(3pi/7))--(cos(pi/14),-sin(pi/14)),dot); draw(arc((0,1),.25,230,310)); MP("A",(0,1),N);MP("B",(cos(pi/14),-sin(pi/14)),E);MP("C",(-cos(pi/14),-sin(pi/14)),W);MP("D",(0,-1/cos(3pi/7)),S); MP("x",(0,.8),S); [/asy]$

An acute isosceles triangle, $ABC$ , is inscribed in a circle. Through $B$ and $C$ , tangents to the circle are drawn, meeting at point $D$ . If $\angle{ABC}=\angle{ACB}=2\angle{D}$ and $x$ is the radian measure of $\angle{A}$ , then $x=$

$\text{(A) } \frac{3\pi}{7}\quad \text{(B) } \frac{4\pi}{9}\quad \text{(C) } \frac{5\pi}{11}\quad \text{(D) } \frac{6\pi}{13}\quad \text{(E) } \frac{7\pi}{15}$

Solution

Problem 15

Four whole numbers, when added three at a time, give the sums $180,197,208$ and $222$ . What is the largest of the four numbers?

$\text{(A) } 77\quad \text{(B) } 83\quad \text{(C) } 89\quad \text{(D) } 95\quad \text{(E) cannot be determined from the given information}$

Solution

Problem 16

At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If $13$ married couples attended, how many handshakes were there among these $26$ people?

$\text{(A) } 78\quad \text{(B) } 185\quad \text{(C) } 234\quad \text{(D) } 312\quad \text{(E) } 325$

Solution

Problem 17

How many of the numbers, $100,101,\cdots,999$ have three different digits in increasing order or in decreasing order?

$\text{(A) } 120\quad \text{(B) } 168\quad \text{(C) } 204\quad \text{(D) } 216\quad \text{(E) } 240$

Solution

Problem 18

First $a$ is chosen at random from the set $\{1,2,3,\cdots,99,100\}$ , and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is

$\text{(A) } \frac{1}{16}\quad \text{(B) } \frac{1}{8}\quad \text{(C) } \frac{3}{16}\quad \text{(D) } \frac{1}{5}\quad \text{(E) } \frac{1}{4}$

Solution

Problem 19

For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ $\underline{not}$ in lowest terms?

$\text{(A) } 0\quad \text{(B) } 86\quad \text{(C) } 90\quad \text{(D) } 104\quad \text{(E) } 105$

Solution

Problem 20

$[asy] draw((0,0)--(7,4.2)--(10,0)--(3,-5)--cycle,dot); draw((0,0)--(3,0)--(7,0)--(10,0),dot); draw((3,-5)--(3,0)--(7,0)--(7,4.2),dot); draw((3/sqrt(34),-5/sqrt(34))--(3/sqrt(34)+1/sqrt(1.36),-5/sqrt(34)+.6/sqrt(1.36))--(1/sqrt(1.36),.6/sqrt(1.36)),black+linewidth(.5)); draw((10-7/sqrt(74),0-5/sqrt(74))--(10-7/sqrt(74)-5/sqrt(74),0-5/sqrt(74)+7/sqrt(74))--(10-5/sqrt(74),7/sqrt(74)),black+linewidth(.5)); draw((3,-1)--(4,-1)--(4,0),black+linewidth(.5)); draw((6,0)--(6,1)--(7,1),black+linewidth(.5)); MP("A",(0,0),W);MP("B",(7,4.2),N);MP("C",(10,0),E);MP("D",(3,-5),S);MP("E",(3,0),N);MP("F",(7,0),S); [/asy]$

In the figure $ABCD$ is a quadrilateral with right angles at $A$ and $C$ . Points $E$ and $F$ are on $\overline{AC}$ , and $\overline{DE}$ and $\overline{BF}$ are perpendicular to $\overline{AC}$ . If $AE=3, DE=5,$ and $CE=7$ , then $BF=$

$\text{(A) } 3.6\quad \text{(B) } 4\quad \text{(C) } 4.2\quad \text{(D) } 4.5\quad \text{(E) } 5$

Solution

Problem 21

Consider a pyramid $P-ABCD$ whose base $ABCD$ is square and whose vertex $P$ is equidistant from $A,B,C$ and $D$ . If $AB=1$ and $\angle{APB}=2\theta$ , then the volume of the pyramid is

$\text{(A) } \frac{\sin(\theta)}{6}\quad \text{(B) } \frac{\cot(\theta)}{6}\quad \text{(C) } \frac{1}{6\sin(\theta)}\quad \text{(D) } \frac{1-\sin(2\theta)}{6}\quad \text{(E) } \frac{\sqrt{\cos(2\theta)}}{6\sin(\theta)}$

Solution

Problem 22

If the six solutions of $x^6=-64$ are written in the form $a+bi$ , where $a$ and $b$ are real, then the product of those solutions with $a>0$ is

$\text{(A) } -2\quad \text{(B) } 0\quad \text{(C) } 2i\quad \text{(D) } 4\quad \text{(E) } 16$

Solution

Problem 23

If $x,y>0, \log_y(x)+\log_x(y)=\frac{10}{3} \text{ and } xy=144,\text{ then }\frac{x+y}{2}=$

$\text{(A) } 12\sqrt{2}\quad \text{(B) } 13\sqrt{3}\quad \text{(C) } 24\quad \text{(D) } 30\quad \text{(E) } 36$

Solution

Problem 24

All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?

$\begin{tabular}[t]{|c|c|c|c|} \multicolumn{4}{c}{Average Scores}\\\hline Category&Adams&Baker&Adams\&Baker\\\hline Boys&71&81&79\\ Girls&76&90&?\\ Boys\&Girls&74&84& \\\hline \end{tabular}$

$\text{(A) } 81\quad \text{(B) } 82\quad \text{(C) } 83\quad \text{(D) } 84\quad \text{(E) } 85$

Solution

Problem 25

Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere?

$\text{(A) } 1-\frac{\sqrt{3}}{2}\quad \text{(B) } \frac{2\sqrt{3}-3}{2}\quad \text{(C) } \frac{\sqrt{2}}{6}\quad \text{(D) } \frac{1}{4}\quad \text{(E) } \frac{\sqrt{3}(2-\sqrt(2))}{4}$

Solution

Problem 26

Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The average announced by each person was (in order around the circle) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (NOT the original number the person picked). The number picked by the person who announced the average 6 was

$\text{(A) } 1\quad \text{(B) } 5\quad \text{(C) } 6\quad \text{(D) } 10\quad \text{(E) not uniquely determined from the given information}$

Solution

Problem 27

Which of these triples could $\underline{not}$ be the lengths of the three altitudes of a triangle?

$\text{(A) } 1,\sqrt{3},2\quad \text{(B) } 3,4,5\quad \text{(C) } 5,12,13\quad \text{(D) } 7,8,\sqrt{113}\quad \text{(E) } 8,15,17$

Solution

Problem 28

A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$ . Find $|x-y|$ .

$\text{(A) } 12\quad \text{(B) } 13\quad \text{(C) } 14\quad \text{(D) } 15\quad \text{(E) } 16$

Solution

Problem 29

A subset of the integers $1,2,\cdots,100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?

$\text{(A) } 50\quad \text{(B) } 66\quad \text{(C) } 67\quad \text{(D) } 76\quad \text{(E) } 78$

Solution

Problem 30

If $R_n=\frac{1}{2}(a^n+b^n)$ where $a=3+2\sqrt{2}$ and $b=3-2\sqrt{2}$ , and $n=0,1,2,\cdots,$ then $R_{12345}$ is an integer. Its units digit is

$\text{(A) } 1\quad \text{(B) } 3\quad \text{(C) } 5\quad \text{(D) } 7\quad \text{(E) } 9$

Solution

1990 AHSME (Problems • Answer Key • Resources)
Preceded by 1989 AHSME	Followed by 1991 AHSME
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
All AHSME Problems and Solutions

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1990 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

See also