# 1998 AJHSME Problems

 1998 AJHSME (Answer Key)Printable versions: Wiki • AoPS Resources • PDF Instructions This is a 25-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 ? points for each correct answer, ? points for each problem left unanswered, and ? 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 ? 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

## Problem 1

For $x=7$, which of the following is the smallest? $\text{(A)}\ \dfrac{6}{x} \qquad \text{(B)}\ \dfrac{6}{x+1} \qquad \text{(C)}\ \dfrac{6}{x-1} \qquad \text{(D)}\ \dfrac{x}{6} \qquad \text{(E)}\ \dfrac{x+1}{6}$

## Problem 2

If $\begin{tabular}{r|l}a&b \\ \hline c&d\end{tabular} = \text{a}\cdot \text{d} - \text{b}\cdot \text{c}$, what is the value of $\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}$? $\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2$

## Problem 3 $\dfrac{\dfrac{3}{8} + \dfrac{7}{8}}{\dfrac{4}{5}} =$ $\text{(A)}\ 1 \qquad \text{(B)} \dfrac{25}{16} \qquad \text{(C)}\ 2 \qquad \text{(D)}\ \dfrac{43}{20} \qquad \text{(E)}\ \dfrac{47}{16}$

## Problem 4

How many triangles are in this figure? (Some triangles may overlap other triangles.) $[asy] draw((0,0)--(42,0)--(14,21)--cycle); draw((14,21)--(18,0)--(30,9)); [/asy]$ $\text{(A)}\ 9 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 5$

## Problem 5

Which of the following numbers is largest? $\text{(A)}\ 9.12344 \qquad \text{(B)}\ 9.123\overline{4} \qquad \text{(C)}\ 9.12\overline{34} \qquad \text{(D)}\ 9.1\overline{234} \qquad \text{(E)}\ 9.\overline{1234}$

## Problem 6

Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is $[asy] for(int a=0; a<4; ++a) { for(int b=0; b<4; ++b) { dot((a,b)); } } draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle); [/asy]$ $\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

## Problem 7 $100\times 19.98\times 1.998\times 1000=$ $\text{(A)}\ (1.998)^2 \qquad \text{(B)}\ (19.98)^2 \qquad \text{(C)}\ (199.8)^2 \qquad \text{(D)}\ (1998)^2 \qquad \text{(E)}\ (19980)^2$

## Problem 8

A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days? $\text{(A)}\ 140 \qquad \text{(B)}\ 170 \qquad \text{(C)}\ 185 \qquad \text{(D)}\ 198.5 \qquad \text{(E)}\ 199.85$

## Problem 9

For a sale, a store owner reduces the price of a 10 scarf by $20\%$. Later the price is lowered again, this time by one-half the reduced price. The price is now $\text{(A)}\ 2.00\text{ dollars} \qquad \text{(B)}\ 3.75\text{ dollars} \qquad \text{(C)}\ 4.00\text{ dollars} \qquad \text{(D)}\ 4.90\text{ dollars} \qquad \text{(E)}\ 6.40\text{ dollars}$

## Problem 10

Each of the letters $\text{W}$, $\text{X}$, $\text{Y}$, and $\text{Z}$ represents a different integer in the set $\{ 1,2,3,4\}$, but not necessarily in that order. If $\dfrac{\text{W}}{\text{X}} - \dfrac{\text{Y}}{\text{Z}}=1$, then the sum of $\text{W}$ and $\text{Y}$ is $\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$

## Problem 11

Harry has 3 sisters and 5 brothers. His sister Harriet has $\text{S}$ sisters and $\text{B}$ brothers. What is the product of $\text{S}$ and $\text{B}$? $\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 18$

## Problem 12 $2\left(1-\dfrac{1}{2}\right) + 3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + \cdots + 10\left(1-\dfrac{1}{10}\right)=$ $\text{(A)}\ 45 \qquad \text{(B)}\ 49 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55$

## Problem 13

What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale) $[asy] draw((0,0)--(0,4)--(4,4)--(4,0)--cycle); draw((0,0)--(4,4)); draw((0,4)--(3,1)--(3,3)); draw((1,1)--(2,0)--(4,2)); fill((1,1)--(2,0)--(3,1)--(2,2)--cycle,black); [/asy]$ $\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{1}{7} \qquad \text{(C)}\ \dfrac{1}{8} \qquad \text{(D)}\ \dfrac{1}{12} \qquad \text{(E)}\ \dfrac{1}{16}$

## Problem 14

At Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club? $\text{(A)}\ 12 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 40$

## Don't Crowd the Isles

Problems 15, 16, and 17 all refer to the following:

In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles.

### Problem 15

Estimate the population of Nisos in the year 2050. $\text{(A)}\ 600 \qquad \text{(B)}\ 800 \qquad \text{(C)}\ 1000 \qquad \text{(D)}\ 2000 \qquad \text{(E)}\ 3000$

### Problem 16

Estimate the year in which the population of Nisos will be approximately 6,000. $\text{(A)}\ 2050 \qquad \text{(B)}\ 2075 \qquad \text{(C)}\ 2100 \qquad \text{(D)}\ 2125 \qquad \text{(E)}\ 2150$

### Problem 17

In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support? $\text{(A)}\ 50\text{ yrs.} \qquad \text{(B)}\ 75\text{ yrs.} \qquad \text{(C)}\ 100\text{ yrs.} \qquad \text{(D)}\ 125\text{ yrs.} \qquad \text{(E)}\ 150\text{ yrs.}$

## Problem 18

As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded? $[asy] draw((2,0)--(2,1)--(4,1)--(4,0)--cycle); draw(circle((2.25,.75),.225)); draw((2.05,.95)--(2.45,.55)); draw((2.45,.95)--(2.05,.55)); draw((0,2)--(4,2)--(4,3)--(0,3)--cycle); draw((2,2)--(2,3),dashed); draw((1.3,2.1)..(2,2.3)..(2.7,2.1),EndArrow); draw((1.3,3.1)..(2,3.3)..(2.7,3.1),EndArrow); draw((0,4)--(4,4)--(4,6)--(0,6)--cycle); draw((0,5)--(4,5),dashed); draw((-.1,4.3)..(-.3,5)..(-.1,5.7),EndArrow); draw((3.9,4.3)..(3.7,5)..(3.9,5.7),EndArrow); [/asy]$ $[asy] unitsize(5); draw((0,0)--(16,0)--(16,8)--(0,8)--cycle); draw((0,4)--(16,4),dashed); draw((8,0)--(8,8),dashed); draw(circle((1,3),.9)); draw(circle((7,7),.9)); draw(circle((15,5),.9)); draw(circle((9,1),.9)); draw((24,0)--(40,0)--(40,8)--(24,8)--cycle); draw((24,4)--(40,4),dashed); draw((32,0)--(32,8),dashed); draw(circle((31,1),.9)); draw(circle((33,1),.9)); draw(circle((31,7),.9)); draw(circle((33,7),.9)); draw((48,0)--(64,0)--(64,8)--(48,8)--cycle); draw((48,4)--(64,4),dashed); draw((56,0)--(56,8),dashed); draw(circle((49,1),.9)); draw(circle((49,7),.9)); draw(circle((63,1),.9)); draw(circle((63,7),.9)); draw((72,0)--(88,0)--(88,8)--(72,8)--cycle); draw((72,4)--(88,4),dashed); draw((80,0)--(80,8),dashed); draw(circle((79,3),.9)); draw(circle((79,5),.9)); draw(circle((81,3),.9)); draw(circle((81,5),.9)); draw((96,0)--(112,0)--(112,8)--(96,8)--cycle); draw((96,4)--(112,4),dashed); draw((104,0)--(104,8),dashed); draw(circle((97,3),.9)); draw(circle((97,5),.9)); draw(circle((111,3),.9)); draw(circle((111,5),.9)); label("(A)",(8,10),N); label("(B)",(32,10),N); label("(C)",(56,10),N); label("(D)",(80,10),N); label("(E)",(104,10),N); [/asy]$

## Problem 19

Tamika selects two different numbers at random from the set $\{ 8,9,10 \}$ and adds them. Carlos takes two different numbers at random from the set $\{3, 5, 6\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? $\text{(A)}\ \dfrac{4}{9} \qquad \text{(B)}\ \dfrac{5}{9} \qquad \text{(C)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}$

## Problem 20

Let $PQRS$ be a square piece of paper. $P$ is folded onto $R$ and then $Q$ is folded onto $S$. The area of the resulting figure is 9 square inches. Find the perimeter of square $PQRS$. $[asy] draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); label("P",(0,2),SE); label("Q",(2,2),SW); label("R",(2,0),NW); label("S",(0,0),NE); [/asy]$ $\text{(A)}\ 9 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 36$

## Problem 21

A $4\times 4\times 4$ cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box? $\text{(A)}\ 48 \qquad \text{(B)}\ 52 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80$

## Problem 22

Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.

Rule 1: If the integer is less than 10, multiply it by 9.

Rule 2: If the integer is even and greater than 9, divide it by 2.

Rule 3: If the integer is odd and greater than 9, subtract 5 from it.

A sample sequence: $23, 18, 9, 81, 76, \ldots .$

Find the $98^\text{th}$ term of the sequence that begins $98, 49, \ldots .$ $\text{(A)}\ 6 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 22 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 54$

## Problem 23

If the pattern in the diagram continues, what fraction of the interior would be shaded in the eighth triangle? $[asy] unitsize(10); draw((0,0)--(12,0)--(6,6sqrt(3))--cycle); draw((15,0)--(27,0)--(21,6sqrt(3))--cycle); fill((21,0)--(18,3sqrt(3))--(24,3sqrt(3))--cycle,black); draw((30,0)--(42,0)--(36,6sqrt(3))--cycle); fill((34,0)--(32,2sqrt(3))--(36,2sqrt(3))--cycle,black); fill((38,0)--(36,2sqrt(3))--(40,2sqrt(3))--cycle,black); fill((36,2sqrt(3))--(34,4sqrt(3))--(38,4sqrt(3))--cycle,black); draw((45,0)--(57,0)--(51,6sqrt(3))--cycle); fill((48,0)--(46.5,1.5sqrt(3))--(49.5,1.5sqrt(3))--cycle,black); fill((51,0)--(49.5,1.5sqrt(3))--(52.5,1.5sqrt(3))--cycle,black); fill((54,0)--(52.5,1.5sqrt(3))--(55.5,1.5sqrt(3))--cycle,black); fill((49.5,1.5sqrt(3))--(48,3sqrt(3))--(51,3sqrt(3))--cycle,black); fill((52.5,1.5sqrt(3))--(51,3sqrt(3))--(54,3sqrt(3))--cycle,black); fill((51,3sqrt(3))--(49.5,4.5sqrt(3))--(52.5,4.5sqrt(3))--cycle,black); [/asy]$ $\text{(A)}\ \dfrac{3}{8} \qquad \text{(B)}\ \dfrac{5}{27} \qquad \text{(C)}\ \dfrac{7}{16} \qquad \text{(D)}\ \dfrac{9}{16} \qquad \text{(E)}\ \dfrac{11}{45}$

## Problem 24

A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result? $[asy] unitsize(24); for(int a = 0; a < 10; ++a) { draw((0,a)--(8,a)); } for (int b = 0; b < 9; ++b) { draw((b,0)--(b,9)); } draw((0,0)--(0,-.5)); draw((1,0)--(1,-1.5)); draw((.5,-1)--(1.5,-1)); draw((2,0)--(2,-.5)); draw((4,0)--(4,-.5)); draw((5,0)--(5,-1.5)); draw((4.5,-1)--(5.5,-1)); draw((6,0)--(6,-.5)); draw((8,0)--(8,-.5)); fill((0,8)--(1,8)--(1,9)--(0,9)--cycle,black); fill((2,8)--(3,8)--(3,9)--(2,9)--cycle,black); fill((5,8)--(6,8)--(6,9)--(5,9)--cycle,black); fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black); fill((6,7)--(7,7)--(7,8)--(6,8)--cycle,black); label("2",(1.5,8.2),N); label("4",(3.5,8.2),N); label("5",(4.5,8.2),N); label("7",(6.5,8.2),N); label("8",(7.5,8.2),N); label("9",(0.5,7.2),N); label("11",(2.5,7.2),N); label("12",(3.5,7.2),N); label("13",(4.5,7.2),N); label("14",(5.5,7.2),N); label("16",(7.5,7.2),N); [/asy]$ $\text{(A)}\ 36 \qquad \text{(B)}\ 64 \qquad \text{(C)}\ 78 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 120$

## Problem 25

Three generous friends, each with some cash, redistribute their money as follows: Amy gives enough money to Jan and Toy to double the amount that each has. Jan then gives enough to Amy and Toy to double their amounts. Finally, Toy gives Amy and Jan enough to double their amounts. If Toy has $36 when they begin and$36 when they end, what is the total amount that all three friends have? $\text{(A)}\ 108\text{ dollars} \qquad \text{(B)}\ 180\text{ dollars} \qquad \text{(C)}\ 216\text{ dollars} \qquad \text{(D)}\ 252\text{ dollars} \qquad \text{(E)}\ 288\text{ dollars}$

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 