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Difference between revisions of "1989 AJHSME Problems"
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{{AJHSME Problems
 +
|year = 1989
 +
}}
 +
 
== Problem 1 ==
 
== Problem 1 ==
  
Line 89: Line 93:
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
 +
Which of the five "T-like shapes" would be symmetric to the one shown with respect to the dashed line?
 +
 +
<asy>
 +
unitsize(48);
 +
for (int a=0; a<3; ++a)
 +
{
 +
  fill((2a+1,1)--(2a+.8,1)--(2a+.8,.8)--(2a+1,.8)--cycle,black);
 +
}
 +
draw((.8,1)--(0,1)--(0,0)--(1,0)--(1,.8));
 +
draw((2.8,1)--(2,1)--(2,0)--(3,0)--(3,.8));
 +
draw((4.8,1)--(4,1)--(4,0)--(5,0)--(5,.8));
 +
draw((.2,.4)--(.6,.8),linewidth(1)); draw((.4,.6)--(.8,.2),linewidth(1));
 +
draw((2.4,.8)--(2.8,.4),linewidth(1)); draw((2.6,.6)--(2.2,.2),linewidth(1));
 +
draw((4.4,.2)--(4.8,.6),linewidth(1)); draw((4.6,.4)--(4.2,.8),linewidth(1));
 +
draw((7,.2)--(7,1)--(6,1)--(6,0)--(6.8,0)); fill((6.8,0)--(7,0)--(7,.2)--(6.8,.2)--cycle,black);
 +
draw((6.2,.6)--(6.6,.2),linewidth(1)); draw((6.4,.4)--(6.8,.8),linewidth(1));
 +
draw((8,.8)--(8,0)--(9,0)--(9,1)--(8.2,1)); fill((8,1)--(8,.8)--(8.2,.8)--(8.2,1)--cycle,black);
 +
draw((8.4,.8)--(8.8,.8),linewidth(1)); draw((8.6,.8)--(8.6,.2),linewidth(1));
 +
draw((6,1.2)--(6,1.4)); draw((6,1.6)--(6,1.8)); draw((6,2)--(6,2.2)); draw((6,2.4)--(6,2.6));
 +
draw((6.4,2.2)--(6.4,1.4)--(7.4,1.4)--(7.4,2.4)--(6.6,2.4)); fill((6.4,2.4)--(6.4,2.2)--(6.6,2.2)--(6.6,2.4)--cycle,black);
 +
draw((6.6,1.8)--(7,2.2),linewidth(1)); draw((6.8,2)--(7.2,1.6),linewidth(1));
 +
label("(A)",(0,1),W); label("(B)",(2,1),W); label("(C)",(4,1),W);
 +
label("(D)",(6,1),W); label("(E)",(8,1),W);
 +
</asy>
  
 
[[1989 AJHSME Problems/Problem 11|Solution]]
 
[[1989 AJHSME Problems/Problem 11|Solution]]
Line 122: Line 151:
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
 +
The area of the shaded region <math>\text{BEDC}</math> in parallelogram <math>\text{ABCD}</math> is
 +
 +
<asy>
 +
unitsize(10);
 +
pair A,B,C,D,E;
 +
A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0);
 +
draw(A--B--C--D--cycle);
 +
fill(B--E--D--C--cycle,gray);
 +
label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,S);
 +
label("$10$",(9,8),N); label("$6$",(7,0),S); label("$8$",(4,4),W);
 +
draw((3,0)--(3,1)--(4,1));
 +
</asy>
 +
 +
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80</math>
  
 
[[1989 AJHSME Problems/Problem 15|Solution]]
 
[[1989 AJHSME Problems/Problem 15|Solution]]
Line 143: Line 187:
 
== Problem 18 ==
 
== Problem 18 ==
  
−
Many calculators have a reciprocal key <math>\boxed{\frac{1}{x}}</math> that replaces the current number displayed with its reciprocal.  For example, if the display is <math>\boxed{00004}</math> and the <math>\boxed{\frac{1}{x}}</math> key is depressed, then the display becomes <math>\boxed{000.25}</math>.  If <math>\boxed{00032}</math> is currently displayed, what is the fewest number of times you must depress the <math>\boxed{\frac{1}{x}}</math> key so the display again reads <math>\boxed{00032}</math>?
+
Many calculators have a reciprocal key <math>\boxed{\frac{1}{x}}</math> that replaces the current number displayed with its reciprocal.  For example, if the display is <math>\boxed{00004}</math> and the <math>\boxed{\frac{1}{x}}</math> key is pressed, then the display becomes <math>\boxed{000.25}</math>.  If <math>\boxed{00032}</math> is currently displayed, what is the fewest positive number of times you must depress the <math>\boxed{\frac{1}{x}}</math> key so the display again reads <math>\boxed{00032}</math>?
  
 
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5</math>
 
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5</math>
Line 150: Line 194:
  
 
== Problem 19 ==
 
== Problem 19 ==
 +
 +
The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during <math>1988</math>.  For example, about <math>.5</math> million had been spent by the beginning of February and approximately <math>2</math> million by the end of April.  Approximately how many millions of dollars were spent during the summer months of June, July, and August?
 +
 +
<math>\text{(A)}\ 1.5 \qquad \text{(B)}\ 2.5 \qquad \text{(C)}\ 3.5 \qquad \text{(D)}\ 4.5 \qquad \text{(E)}\ 5.5</math>
 +
 +
<asy>
 +
unitsize(18);
 +
for (int a=1; a<13; ++a)
 +
{
 +
  draw((a,0)--(a,.5));
 +
}
 +
for (int b=1; b<6; ++b)
 +
{
 +
  draw((-.5,2b)--(0,2b));
 +
}
 +
draw((0,0)--(0,12));
 +
draw((0,0)--(14,0));
 +
draw((0,0)--(1,.9)--(2,1.9)--(3,2.6)--(4,4.3)--(5,4.5)--(6,5.7)--(7,8.2)--(8,9.4)--(9,9.8)--(10,10.1)--(11,10.2)--(12,10.5));
 +
label("J",(.5,0),S); label("F",(1.5,0),S); label("M",(2.5,0),S); label("A",(3.5,0),S);
 +
label("M",(4.5,0),S); label("J",(5.5,0),S); label("J",(6.5,0),S); label("A",(7.5,0),S);
 +
label("S",(8.5,0),S); label("O",(9.5,0),S); label("N",(10.5,0),S); label("D",(11.5,0),S);
 +
label("month F=February",(16,0),S);
 +
label("$1$",(-.6,2),W); label("$2$",(-.6,4),W); label("$3$",(-.6,6),W);
 +
label("$4$",(-.6,8),W); label("$5$",(-.6,10),W);
 +
label("dollars in millions",(0,11.9),N);
 +
</asy>
  
 
[[1989 AJHSME Problems/Problem 19|Solution]]
 
[[1989 AJHSME Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
 +
The figure may be folded along the lines shown to form a number cube.  Three number faces come together at each corner of the cube.  What is the largest sum of three numbers whose faces come together at a corner?
 +
 +
<asy>
 +
draw((0,0)--(0,1)--(1,1)--(1,2)--(2,2)--(2,1)--(4,1)--(4,0)--(2,0)--(2,-1)--(1,-1)--(1,0)--cycle);
 +
draw((1,0)--(1,1)--(2,1)--(2,0)--cycle); draw((3,1)--(3,0));
 +
label("$1$",(1.5,1.25),N); label("$2$",(1.5,.25),N); label("$3$",(1.5,-.75),N);
 +
label("$4$",(2.5,.25),N); label("$5$",(3.5,.25),N); label("$6$",(.5,.25),N);
 +
</asy>
 +
 +
<math>\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15</math>
  
 
[[1989 AJHSME Problems/Problem 20|Solution]]
 
[[1989 AJHSME Problems/Problem 20|Solution]]
Line 166: Line 247:
  
 
== Problem 22 ==
 
== Problem 22 ==
 +
 +
The letters <math>\text{A}</math>, <math>\text{J}</math>, <math>\text{H}</math>, <math>\text{S}</math>, <math>\text{M}</math>, <math>\text{E}</math> and the digits <math>1</math>, <math>9</math>, <math>8</math>, <math>9</math> are "cycled" separately as follows and put together in a numbered list:
 +
<cmath>\begin{tabular}[t]{lccc}
 +
& & AJHSME & 1989 \\
 +
& & & \\
 +
1. & & JHSMEA & 9891 \\
 +
2. & & HSMEAJ & 8919 \\
 +
3. & & SMEAJH & 9198 \\
 +
& & ........ &
 +
\end{tabular}</cmath>
 +
 +
What is the number of the line on which <math>\text{AJHSME 1989}</math> will appear for the first time?
 +
 +
<math>\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24</math>
  
 
[[1989 AJHSME Problems/Problem 22|Solution]]
 
[[1989 AJHSME Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
 +
An artist has <math>14</math> cubes, each with an edge of <math>1</math> meter.  She stands them on the ground to form a sculpture as shown.  She then paints the exposed surface of the sculpture.  How many square meters does she paint?
 +
 +
<math>\text{(A)}\ 21 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 33 \qquad \text{(D)}\ 37 \qquad \text{(E)}\ 42</math>
 +
 +
<asy>
 +
draw((0,0)--(2.35,-.15)--(2.44,.81)--(.09,.96)--cycle);
 +
draw((.783333333,-.05)--(.873333333,.91)--(1.135,1.135));
 +
draw((1.566666667,-.1)--(1.656666667,.86)--(1.89,1.1));
 +
draw((2.35,-.15)--(4.3,1.5)--(4.39,2.46)--(2.44,.81));
 +
draw((3,.4)--(3.09,1.36)--(2.61,1.4));
 +
draw((3.65,.95)--(3.74,1.91)--(3.29,1.94));
 +
draw((.09,.96)--(.76,1.49)--(.71,1.17)--(2.2,1.1)--(3.6,2.2)--(3.62,2.52)--(4.39,2.46));
 +
draw((.76,1.49)--(.82,1.96)--(2.28,1.89)--(2.2,1.1));
 +
draw((2.28,1.89)--(3.68,2.99)--(3.62,2.52));
 +
draw((1.455,1.135)--(1.55,1.925)--(1.89,2.26));
 +
draw((2.5,2.48)--(2.98,2.44)--(2.9,1.65));
 +
draw((.82,1.96)--(1.55,2.6)--(1.51,2.3)--(2.2,2.26)--(2.9,2.8)--(2.93,3.05)--(3.68,2.99));
 +
draw((1.55,2.6)--(1.59,3.09)--(2.28,3.05)--(2.2,2.26));
 +
draw((2.28,3.05)--(2.98,3.59)--(2.93,3.05));
 +
draw((1.59,3.09)--(2.29,3.63)--(2.98,3.59));
 +
</asy>
  
 
[[1989 AJHSME Problems/Problem 23|Solution]]
 
[[1989 AJHSME Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
 +
 +
Suppose a square piece of paper is folded in half vertically.  The folded paper is then cut in half along the dashed line.  Three rectangles are formed-a large one and two small ones.  What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
 +
 +
<math>\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}</math>
 +
 +
<asy>
 +
draw((0,0)--(0,8)--(6,8)--(6,0)--cycle);
 +
draw((0,8)--(5,9)--(5,8));
 +
draw((3,-1.5)--(3,10.3),dashed);
 +
draw((0,5.5)..(-.75,4.75)..(0,4));
 +
draw((0,4)--(1.5,4),EndArrow);
 +
</asy>
  
 
[[1989 AJHSME Problems/Problem 24|Solution]]
 
[[1989 AJHSME Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== Problem 25 ==
 +
 +
Every time these two wheels are spun, two numbers are selected by the pointers.  What is the probability that the sum of the two selected numbers is even?
 +
 +
<math>\text{(A)}\ \frac{1}{6} \qquad \text{(B)}\ \frac{3}{7} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{2}{3} \qquad \text{(E)}\ \frac{5}{7}</math>
 +
 +
<asy>
 +
unitsize(36);
 +
draw(circle((-3,0),1));
 +
draw(circle((0,0),1));
 +
draw((0,0)--dir(30)); draw((0,0)--(0,-1)); draw((0,0)--dir(150));
 +
draw((-2.293,.707)--(-3.707,-.707)); draw((-2.293,-.707)--(-3.707,.707));
 +
fill((-2.9,1)--(-2.65,1.25)--(-2.65,1.6)--(-3.35,1.6)--(-3.35,1.25)--(-3.1,1)--cycle,black);
 +
fill((.1,1)--(.35,1.25)--(.35,1.6)--(-.35,1.6)--(-.35,1.25)--(-.1,1)--cycle,black);
 +
label("$5$",(-3,.2),N); label("$3$",(-3.2,0),W); label("$4$",(-3,-.2),S); label("$8$",(-2.8,0),E);
 +
label("$6$",(0,.2),N); label("$9$",(-.2,.1),SW); label("$7$",(.2,.1),SE);
 +
</asy>
  
 
[[1989 AJHSME Problems/Problem 25|Solution]]
 
[[1989 AJHSME Problems/Problem 25|Solution]]
  
 
== See also ==
 
== See also ==
−
 
+
{{AJHSME box|year=1989|before=[[1988 AJHSME Problems|1988 AJHSME]]|after=[[1990 AJHSME Problems|1990 AJHSME]]}}
 
* [[AJHSME]]
 
* [[AJHSME]]
 
* [[AJHSME Problems and Solutions]]
 
* [[AJHSME Problems and Solutions]]
−
* [[1989 AJHSME]]
 
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
 +
 +
{{MAA Notice}}

Latest revision as of 10:55, 22 November 2023

1989 AJHSME (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. 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.
  2. You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers.
  4. Figures are not necessarily drawn to scale.
  5. 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

$(1+11+21+31+41)+(9+19+29+39+49)=$

$\text{(A)}\ 150 \qquad \text{(B)}\ 199 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 249 \qquad \text{(E)}\ 250$

Solution

Problem 2

$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =$

$\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$

Solution

Problem 3

Which of the following numbers is the largest?

$\text{(A)}\ .99 \qquad \text{(B)}\ .9099 \qquad \text{(C)}\ .9 \qquad \text{(D)}\ .909 \qquad \text{(E)}\ .9009$

Solution

Problem 4

Estimate to determine which of the following numbers is closest to $\frac{401}{.205}$.

$\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 200 \qquad \text{(E)}\ 2000$

Solution

Problem 5

$-15+9\times (6\div 3) =$

$\text{(A)}\ -48 \qquad \text{(B)}\ -12 \qquad \text{(C)}\ -3 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 12$

Solution

Problem 6

If the markings on the number line are equally spaced, what is the number $\text{y}$?

[asy] draw((-4,0)--(26,0),Arrows); for(int a=0; a<6; ++a) { draw((4a,-1)--(4a,1)); } label("0",(0,-1),S); label("20",(20,-1),S); label("y",(12,-1),S); [/asy]

$\text{(A)}\ 3 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 16$

Solution

Problem 7

If the value of $20$ quarters and $10$ dimes equals the value of $10$ quarters and $n$ dimes, then $n=$

$\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 45$

Solution

Problem 8

$(2\times 3\times 4)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right) =$

$\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 26$

Solution

Problem 9

There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys?

$\text{(A)}\ 12\% \qquad \text{(B)}\ 20\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 60\% \qquad \text{(E)}\ 66\frac{2}{3}\%$

Solution

Problem 10

What is the number of degrees in the smaller angle between the hour hand and the minute hand on a clock that reads seven o'clock?

$\text{(A)}\ 50^\circ \qquad \text{(B)}\ 120^\circ \qquad \text{(C)}\ 135^\circ \qquad \text{(D)}\ 150^\circ \qquad \text{(E)}\ 165^\circ$

Solution

Problem 11

Which of the five "T-like shapes" would be symmetric to the one shown with respect to the dashed line?

[asy] unitsize(48); for (int a=0; a<3; ++a) { fill((2a+1,1)--(2a+.8,1)--(2a+.8,.8)--(2a+1,.8)--cycle,black); } draw((.8,1)--(0,1)--(0,0)--(1,0)--(1,.8)); draw((2.8,1)--(2,1)--(2,0)--(3,0)--(3,.8)); draw((4.8,1)--(4,1)--(4,0)--(5,0)--(5,.8)); draw((.2,.4)--(.6,.8),linewidth(1)); draw((.4,.6)--(.8,.2),linewidth(1)); draw((2.4,.8)--(2.8,.4),linewidth(1)); draw((2.6,.6)--(2.2,.2),linewidth(1)); draw((4.4,.2)--(4.8,.6),linewidth(1)); draw((4.6,.4)--(4.2,.8),linewidth(1)); draw((7,.2)--(7,1)--(6,1)--(6,0)--(6.8,0)); fill((6.8,0)--(7,0)--(7,.2)--(6.8,.2)--cycle,black); draw((6.2,.6)--(6.6,.2),linewidth(1)); draw((6.4,.4)--(6.8,.8),linewidth(1)); draw((8,.8)--(8,0)--(9,0)--(9,1)--(8.2,1)); fill((8,1)--(8,.8)--(8.2,.8)--(8.2,1)--cycle,black); draw((8.4,.8)--(8.8,.8),linewidth(1)); draw((8.6,.8)--(8.6,.2),linewidth(1)); draw((6,1.2)--(6,1.4)); draw((6,1.6)--(6,1.8)); draw((6,2)--(6,2.2)); draw((6,2.4)--(6,2.6)); draw((6.4,2.2)--(6.4,1.4)--(7.4,1.4)--(7.4,2.4)--(6.6,2.4)); fill((6.4,2.4)--(6.4,2.2)--(6.6,2.2)--(6.6,2.4)--cycle,black); draw((6.6,1.8)--(7,2.2),linewidth(1)); draw((6.8,2)--(7.2,1.6),linewidth(1)); label("(A)",(0,1),W); label("(B)",(2,1),W); label("(C)",(4,1),W); label("(D)",(6,1),W); label("(E)",(8,1),W); [/asy]

Solution

Problem 12

$\frac{1-\frac{1}{3}}{1-\frac{1}{2}} =$

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

Solution

Problem 13

$\frac{9}{7\times 53} =$

$\text{(A)}\ \frac{.9}{.7\times 53} \qquad \text{(B)}\ \frac{.9}{.7\times .53} \qquad \text{(C)}\ \frac{.9}{.7\times 5.3} \qquad \text{(D)}\ \frac{.9}{7\times .53} \qquad \text{(E)}\ \frac{.09}{.07\times .53}$

Solution

Problem 14

When placing each of the digits $2,4,5,6,9$ in exactly one of the boxes of this subtraction problem, what is the smallest difference that is possible?

$\text{(A)}\ 58 \qquad \text{(B)}\ 123 \qquad \text{(C)}\ 149 \qquad \text{(D)}\ 171 \qquad \text{(E)}\ 176$

\[\begin{tabular}[t]{cccc} & \boxed{} & \boxed{} & \boxed{} \\ - & & \boxed{} & \boxed{} \\ \hline \end{tabular}\]

Solution

Problem 15

The area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$ is

[asy] unitsize(10); pair A,B,C,D,E; A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0); draw(A--B--C--D--cycle); fill(B--E--D--C--cycle,gray); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,S); label("$10$",(9,8),N); label("$6$",(7,0),S); label("$8$",(4,4),W); draw((3,0)--(3,1)--(4,1)); [/asy]

$\text{(A)}\ 24 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80$

Solution

Problem 16

In how many ways can $47$ be written as the sum of two primes?

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

Solution

Problem 17

The number $\text{N}$ is between $9$ and $17$. The average of $6$, $10$, and $\text{N}$ could be

$\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$

Solution

Problem 18

Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is pressed, then the display becomes $\boxed{000.25}$. If $\boxed{00032}$ is currently displayed, what is the fewest positive number of times you must depress the $\boxed{\frac{1}{x}}$ key so the display again reads $\boxed{00032}$?

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

Solution

Problem 19

The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during $1988$. For example, about $.5$ million had been spent by the beginning of February and approximately $2$ million by the end of April. Approximately how many millions of dollars were spent during the summer months of June, July, and August?

$\text{(A)}\ 1.5 \qquad \text{(B)}\ 2.5 \qquad \text{(C)}\ 3.5 \qquad \text{(D)}\ 4.5 \qquad \text{(E)}\ 5.5$

[asy] unitsize(18); for (int a=1; a<13; ++a) { draw((a,0)--(a,.5)); } for (int b=1; b<6; ++b) { draw((-.5,2b)--(0,2b)); } draw((0,0)--(0,12)); draw((0,0)--(14,0)); draw((0,0)--(1,.9)--(2,1.9)--(3,2.6)--(4,4.3)--(5,4.5)--(6,5.7)--(7,8.2)--(8,9.4)--(9,9.8)--(10,10.1)--(11,10.2)--(12,10.5)); label("J",(.5,0),S); label("F",(1.5,0),S); label("M",(2.5,0),S); label("A",(3.5,0),S); label("M",(4.5,0),S); label("J",(5.5,0),S); label("J",(6.5,0),S); label("A",(7.5,0),S); label("S",(8.5,0),S); label("O",(9.5,0),S); label("N",(10.5,0),S); label("D",(11.5,0),S); label("month F=February",(16,0),S); label("$1$",(-.6,2),W); label("$2$",(-.6,4),W); label("$3$",(-.6,6),W); label("$4$",(-.6,8),W); label("$5$",(-.6,10),W); label("dollars in millions",(0,11.9),N); [/asy]

Solution

Problem 20

The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner?

[asy] draw((0,0)--(0,1)--(1,1)--(1,2)--(2,2)--(2,1)--(4,1)--(4,0)--(2,0)--(2,-1)--(1,-1)--(1,0)--cycle); draw((1,0)--(1,1)--(2,1)--(2,0)--cycle); draw((3,1)--(3,0)); label("$1$",(1.5,1.25),N); label("$2$",(1.5,.25),N); label("$3$",(1.5,-.75),N); label("$4$",(2.5,.25),N); label("$5$",(3.5,.25),N); label("$6$",(.5,.25),N); [/asy]

$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$

Solution

Problem 21

Jack had a bag of $128$ apples. He sold $25\%$ of them to Jill. Next he sold $25\%$ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?

$\text{(A)}\ 7 \qquad \text{(B)}\ 63 \qquad \text{(C)}\ 65 \qquad \text{(D)}\ 71 \qquad \text{(E)}\ 111$

Solution

Problem 22

The letters $\text{A}$, $\text{J}$, $\text{H}$, $\text{S}$, $\text{M}$, $\text{E}$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list: \[\begin{tabular}[t]{lccc} & & AJHSME & 1989 \\ & & & \\ 1. & & JHSMEA & 9891 \\ 2. & & HSMEAJ & 8919 \\ 3. & & SMEAJH & 9198 \\ & & ........ & \end{tabular}\]

What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time?

$\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24$

Solution

Problem 23

An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint?

$\text{(A)}\ 21 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 33 \qquad \text{(D)}\ 37 \qquad \text{(E)}\ 42$

[asy] draw((0,0)--(2.35,-.15)--(2.44,.81)--(.09,.96)--cycle); draw((.783333333,-.05)--(.873333333,.91)--(1.135,1.135)); draw((1.566666667,-.1)--(1.656666667,.86)--(1.89,1.1)); draw((2.35,-.15)--(4.3,1.5)--(4.39,2.46)--(2.44,.81)); draw((3,.4)--(3.09,1.36)--(2.61,1.4)); draw((3.65,.95)--(3.74,1.91)--(3.29,1.94)); draw((.09,.96)--(.76,1.49)--(.71,1.17)--(2.2,1.1)--(3.6,2.2)--(3.62,2.52)--(4.39,2.46)); draw((.76,1.49)--(.82,1.96)--(2.28,1.89)--(2.2,1.1)); draw((2.28,1.89)--(3.68,2.99)--(3.62,2.52)); draw((1.455,1.135)--(1.55,1.925)--(1.89,2.26)); draw((2.5,2.48)--(2.98,2.44)--(2.9,1.65)); draw((.82,1.96)--(1.55,2.6)--(1.51,2.3)--(2.2,2.26)--(2.9,2.8)--(2.93,3.05)--(3.68,2.99)); draw((1.55,2.6)--(1.59,3.09)--(2.28,3.05)--(2.2,2.26)); draw((2.28,3.05)--(2.98,3.59)--(2.93,3.05)); draw((1.59,3.09)--(2.29,3.63)--(2.98,3.59)); [/asy]

Solution

Problem 24

Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?

$\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}$

[asy] draw((0,0)--(0,8)--(6,8)--(6,0)--cycle); draw((0,8)--(5,9)--(5,8)); draw((3,-1.5)--(3,10.3),dashed); draw((0,5.5)..(-.75,4.75)..(0,4)); draw((0,4)--(1.5,4),EndArrow); [/asy]

Solution

Problem 25

Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even?

$\text{(A)}\ \frac{1}{6} \qquad \text{(B)}\ \frac{3}{7} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{2}{3} \qquad \text{(E)}\ \frac{5}{7}$

[asy] unitsize(36); draw(circle((-3,0),1)); draw(circle((0,0),1)); draw((0,0)--dir(30)); draw((0,0)--(0,-1)); draw((0,0)--dir(150)); draw((-2.293,.707)--(-3.707,-.707)); draw((-2.293,-.707)--(-3.707,.707)); fill((-2.9,1)--(-2.65,1.25)--(-2.65,1.6)--(-3.35,1.6)--(-3.35,1.25)--(-3.1,1)--cycle,black); fill((.1,1)--(.35,1.25)--(.35,1.6)--(-.35,1.6)--(-.35,1.25)--(-.1,1)--cycle,black); label("$5$",(-3,.2),N); label("$3$",(-3.2,0),W); label("$4$",(-3,-.2),S); label("$8$",(-2.8,0),E); label("$6$",(0,.2),N); label("$9$",(-.2,.1),SW); label("$7$",(.2,.1),SE); [/asy]

Solution

See also

1989 AJHSME (Problems • Answer Key • Resources)
Preceded by
1988 AJHSME 		Followed by
1990 AJHSME
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
All AJHSME/AMC 8 Problems and Solutions


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