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Difference between revisions of "1992 AJHSME Problems"
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 +
{{AJHSME Problems
 +
|year = 1992
 +
}}
 
==Problem 1==
 
==Problem 1==
  
Line 25: Line 28:
 
== Problem 4 ==
 
== Problem 4 ==
  
−
During the softball season, Judy had <math>35</math> hits.  Among her hits were <math>1</math> home run, <math>1</math> triple and <math>5</math> doubles.  The rest of her hits were single.  What percent of her hits were single?
+
During the softball season, Judy had <math>35</math> hits.  Among her hits were <math>1</math> home run, <math>1</math> triple and <math>5</math> doubles.  The rest of her hits were singles.  What percent of her hits were singles?
  
 
<math>\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\% </math>
 
<math>\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\% </math>
Line 47: Line 50:
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
 +
Suppose that
 +
<asy>
 +
unitsize(18);
 +
draw((0,0)--(2,0)--(1,sqrt(3))--cycle);
 +
label("$a$",(1,sqrt(3)-0.2),S);
 +
label("$b$",(sqrt(3)/10,0.1),ENE);
 +
label("$c$",(2-sqrt(3)/10,0.1),WNW);
 +
</asy>
 +
means <math>a+b-c</math>.
 +
For example,
 +
<asy>
 +
unitsize(18);
 +
draw((0,0)--(2,0)--(1,sqrt(3))--cycle);
 +
label("$5$",(1,sqrt(3)-0.2),S);
 +
label("$4$",(sqrt(3)/10,0.1),ENE);
 +
label("$6$",(2-sqrt(3)/10,0.1),WNW);
 +
</asy>
 +
is <math>5+4-6 = 3</math>.
 +
Then the sum
 +
<asy>
 +
unitsize(18);
 +
draw((0,0)--(2,0)--(1,sqrt(3))--cycle);
 +
label("$1$",(1,sqrt(3)-0.2),S);
 +
label("$3$",(sqrt(3)/10,0.1),ENE);
 +
label("$4$",(2-sqrt(3)/10,0.1),WNW);
 +
draw((3,0)--(5,0)--(4,sqrt(3))--cycle);
 +
label("$2$",(4,sqrt(3)-0.2),S);
 +
label("$5$",(3+sqrt(3)/10,0.1),ENE);
 +
label("$6$",(5-sqrt(3)/10,0.1),WNW);
 +
label("$+$",(2.5,-0.1),N);
 +
</asy>
 +
is
 +
 +
<math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2</math>
  
 
[[1992 AJHSME Problems/Problem 6|Solution]]
 
[[1992 AJHSME Problems/Problem 6|Solution]]
Line 60: Line 98:
 
== Problem 8 ==
 
== Problem 8 ==
  
−
A store owner bought <math>1500</math> pencils at <dollar/>0.10 each.  If he sells them for <dollar/>0.25 each, how many of them must he sell to make a profit of exactly <dollar/>100.00?
+
A store owner bought <math>1500</math> pencils at <math>\$0.10</math> each.  If he sells them for <math>\$0.25</math> each, how many of them must he sell to make a profit of exactly <math>\$100.00</math>?
  
 
<math>\text{(A)}\ 400 \qquad \text{(B)}\ 667 \qquad \text{(C)}\ 1000 \qquad \text{(D)}\ 1500 \qquad \text{(E)}\ 1900</math>
 
<math>\text{(A)}\ 400 \qquad \text{(B)}\ 667 \qquad \text{(C)}\ 1000 \qquad \text{(D)}\ 1500 \qquad \text{(E)}\ 1900</math>
Line 87: Line 125:
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
 +
An isosceles right triangle with legs of length <math>8</math> is partitioned into <math>16</math> congruent triangles as shown.  The shaded area is
 +
 +
<asy>
 +
for (int a=0; a <= 3; ++a)
 +
{
 +
    for (int b=0; b <= 3-a; ++b)
 +
    {
 +
        fill((a,b)--(a,b+1)--(a+1,b)--cycle,grey);
 +
    }
 +
}
 +
for (int c=0; c <= 3; ++c)
 +
{
 +
    draw((c,0)--(c,4-c),linewidth(1));
 +
    draw((0,c)--(4-c,c),linewidth(1));
 +
    draw((c+1,0)--(0,c+1),linewidth(1));
 +
}
 +
 +
label("$8$",(2,0),S);
 +
label("$8$",(0,2),W);
 +
</asy>
 +
 +
<math>\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 64</math>
  
 
[[1992 AJHSME Problems/Problem 10|Solution]]
 
[[1992 AJHSME Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
 +
The bar graph shows the results of a survey on color preferences.  What percent preferred blue?
 +
 +
<asy>
 +
for (int a = 1; a <= 6; ++a)
 +
{
 +
    draw((-1.5,4*a)--(1.5,4*a));
 +
}
 +
draw((0,28)--(0,0)--(32,0));
 +
draw((3,0)--(3,20)--(6,20)--(6,0));
 +
draw((9,0)--(9,24)--(12,24)--(12,0));
 +
draw((15,0)--(15,16)--(18,16)--(18,0));
 +
draw((21,0)--(21,24)--(24,24)--(24,0));
 +
draw((27,0)--(27,16)--(30,16)--(30,0));
 +
 +
label("$20$",(-1.5,8),W);
 +
label("$40$",(-1.5,16),W);
 +
label("$60$",(-1.5,24),W);
 +
 +
label("$\textbf{COLOR SURVEY}$",(16,26),N);
 +
label("$\textbf{F}$",(-6,25),W);
 +
label("$\textbf{r}$",(-6.75,22.4),W);
 +
label("$\textbf{e}$",(-6.75,19.8),W);
 +
label("$\textbf{q}$",(-6.75,17.2),W);
 +
label("$\textbf{u}$",(-6.75,15),W);
 +
label("$\textbf{e}$",(-6.75,12.4),W);
 +
label("$\textbf{n}$",(-6.75,9.8),W);
 +
label("$\textbf{c}$",(-6.75,7.2),W);
 +
label("$\textbf{y}$",(-6.75,4.6),W);
 +
 +
label("D",(4.5,.2),N);
 +
label("E",(4.5,3),N);
 +
label("R",(4.5,5.8),N);
 +
 +
label("E",(10.5,.2),N);
 +
label("U",(10.5,3),N);
 +
label("L",(10.5,5.8),N);
 +
label("B",(10.5,8.6),N);
 +
 +
label("N",(16.5,.2),N);
 +
label("W",(16.5,3),N);
 +
label("O",(16.5,5.8),N);
 +
label("R",(16.5,8.6),N);
 +
label("B",(16.5,11.4),N);
 +
 +
label("K",(22.5,.2),N);
 +
label("N",(22.5,3),N);
 +
label("I",(22.5,5.8),N);
 +
label("P",(22.5,8.6),N);
 +
 +
label("N",(28.5,.2),N);
 +
label("E",(28.5,3),N);
 +
label("E",(28.5,5.8),N);
 +
label("R",(28.5,8.6),N);
 +
label("G",(28.5,11.4),N);
 +
</asy>
 +
 +
<math>\text{(A)}\ 20\% \qquad \text{(B)}\ 24\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 36\% \qquad \text{(E)}\ 42\% </math>
  
 
[[1992 AJHSME Problems/Problem 11|Solution]]
 
[[1992 AJHSME Problems/Problem 11|Solution]]
Line 129: Line 248:
  
 
== Problem 16 ==
 
== Problem 16 ==
 +
 +
<asy>
 +
draw(ellipse((0,-5),10,3));
 +
fill((-10,-5)--(10,-5)--(10,5)--(-10,5)--cycle,white);
 +
draw(ellipse((0,0),10,3));
 +
draw((10,0)--(10,-5));
 +
draw((-10,0)--(-10,-5));
 +
 +
draw((0,0)--(7,-3*sqrt(51)/10));
 +
label("10",(7/2,-3*sqrt(51)/20),NE);
 +
label("5",(-10,-3),E);
 +
</asy>
 +
 +
Which cylinder has twice the volume of the cylinder shown above?
 +
 +
<asy>
 +
unitsize(4);
 +
 +
draw(ellipse((0,-5),20,6));
 +
fill((-20,-5)--(20,-5)--(20,5)--(-20,5)--cycle,white);
 +
draw(ellipse((0,0),20,6));
 +
draw((20,0)--(20,-5));
 +
draw((-20,0)--(-20,-5));
 +
draw((0,0)--(14,-3*sqrt(51)/5));
 +
label("20",(7,-3*sqrt(51)/10),NE);
 +
label("5",(-20,-4),E);
 +
label("(A)",(0,6),N);
 +
 +
draw(ellipse((31,-7),10,3));
 +
fill((21,-7)--(41,-7)--(41,7)--(21,7)--cycle,white);
 +
draw(ellipse((31,3),10,3));
 +
draw((41,3)--(41,-7));
 +
draw((21,3)--(21,-7));
 +
draw((31,3)--(38,3-3*sqrt(51)/10));
 +
label("10",(34.5,3-3*sqrt(51)/20),NE);
 +
label("10",(21,-4),E);
 +
label("(B)",(31,6),N);
 +
 +
draw(ellipse((47,-15.5),5,3/2));
 +
fill((42,-15.5)--(42,-15.5)--(42,15.5)--(42,15.5)--cycle,white);
 +
draw(ellipse((47,4.5),5,3/2));
 +
draw((42,4.5)--(42,-15.5));
 +
draw((52,4.5)--(52,-15.5));
 +
draw((47,4.5)--(50.5,4.5-3*sqrt(51)/20));
 +
label("5",(48.75,4.5-3*sqrt(51)/40),NE);
 +
label("10",(42,-6),E);
 +
label("(C)",(47,6),N);
 +
 +
draw(ellipse((73,-10),20,6));
 +
fill((53,-10)--(93,-10)--(93,5)--(53,5)--cycle,white);
 +
draw(ellipse((73,0),20,6));
 +
draw((53,0)--(53,-10));
 +
draw((93,0)--(93,-10));
 +
draw((73,0)--(87,-3*sqrt(51)/5));
 +
label("20",(80,-3*sqrt(51)/10),NE);
 +
label("10",(53,-6),E);
 +
label("(D)",(73,6),N);
 +
</asy>
 +
 +
<math>\text{(E)}\ \text{None of the above}</math>
  
 
[[1992 AJHSME Problems/Problem 16|Solution]]
 
[[1992 AJHSME Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== Problem 17 ==
 +
 +
The sides of a triangle have lengths <math>6.5</math>, <math>10</math>, and <math>s</math>, where <math>s</math> is a whole number.  What is the smallest possible value of <math>s</math>?
 +
 +
<asy>
 +
pair A,B,C;
 +
A=origin; B=(10,0); C=6.5*dir(15);
 +
dot(A); dot(B); dot(C);
 +
draw(B--A--C);
 +
draw(B--C,dashed);
 +
label("$6.5$",3.25*dir(15),NNW);
 +
label("$10$",(5,0),S);
 +
label("$s$",(8,1),NE);
 +
</asy>
 +
 +
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7</math>
  
 
[[1992 AJHSME Problems/Problem 17|Solution]]
 
[[1992 AJHSME Problems/Problem 17|Solution]]
Line 153: Line 347:
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
 +
Which pattern of identical squares could NOT be folded along the lines shown to form a cube?
 +
 +
<asy>
 +
unitsize(12);
 +
 +
draw((0,0)--(0,-1)--(1,-1)--(1,-2)--(2,-2)--(2,-3)--(4,-3)--(4,-2)--(3,-2)--(3,-1)--(2,-1)--(2,0)--cycle);
 +
draw((1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3));
 +
 +
draw((7,0)--(8,0)--(8,-1)--(11,-1)--(11,-2)--(8,-2)--(8,-3)--(7,-3)--cycle);
 +
draw((7,-1)--(8,-1)--(8,-2)--(7,-2));
 +
draw((9,-1)--(9,-2));
 +
draw((10,-1)--(10,-2));
 +
 +
draw((14,-1)--(15,-1)--(15,0)--(16,0)--(16,-1)--(18,-1)--(18,-2)--(17,-2)--(17,-3)--(16,-3)--(16,-2)--(14,-2)--cycle);
 +
draw((15,-2)--(15,-1)--(16,-1)--(16,-2)--(17,-2)--(17,-1));
 +
 +
draw((21,-1)--(22,-1)--(22,0)--(23,0)--(23,-2)--(25,-2)--(25,-3)--(22,-3)--(22,-2)--(21,-2)--cycle);
 +
draw((23,-1)--(22,-1)--(22,-2)--(23,-2)--(23,-3));
 +
draw((24,-2)--(24,-3));
 +
 +
draw((28,-1)--(31,-1)--(31,0)--(32,0)--(32,-2)--(31,-2)--(31,-3)--(30,-3)--(30,-2)--(28,-2)--cycle);
 +
draw((32,-1)--(31,-1)--(31,-2)--(30,-2)--(30,-1));
 +
draw((29,-1)--(29,-2));
 +
 +
label("(A)",(0,-0.5),W);
 +
label("(B)",(7,-0.5),W);
 +
label("(C)",(14,-0.5),W);
 +
label("(D)",(21,-0.5),W);
 +
label("(E)",(28,-0.5),W);
 +
</asy>
  
 
[[1992 AJHSME Problems/Problem 20|Solution]]
 
[[1992 AJHSME Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
 +
Northside's Drum and Bugle Corps raised money for a trip.  The drummers and bugle players kept separate sales records.  According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?
 +
 +
<asy>
 +
unitsize(12);
 +
 +
fill((2,0)--(2,9)--(3,9)--(3,0)--cycle,lightgray);
 +
draw((3,0)--(3,9)--(2,9)--(2,0));
 +
draw((2,7)--(1,7)--(1,0));
 +
draw((2,8)--(3,8));
 +
draw((2,7)--(3,7));
 +
for (int a = 1; a <= 6; ++a)
 +
{
 +
    draw((1,a)--(3,a));
 +
}
 +
 +
fill((5,0)--(5,3)--(6,3)--(6,0)--cycle,lightgray);
 +
draw((6,0)--(6,3)--(5,3)--(5,0));
 +
draw((5,3)--(5,5)--(4,5)--(4,0));
 +
draw((4,4)--(5,4));
 +
draw((4,3)--(5,3));
 +
draw((4,2)--(6,2));
 +
draw((4,1)--(6,1));
 +
 +
fill((8,0)--(8,6)--(9,6)--(9,0)--cycle,lightgray);
 +
draw((9,0)--(9,6)--(8,6)--(8,0));
 +
draw((8,6)--(8,9)--(7,9)--(7,0));
 +
draw((7,8)--(8,8));
 +
draw((7,7)--(8,7));
 +
draw((7,6)--(8,6));
 +
for (int a = 1; a <= 5; ++a)
 +
{
 +
    draw((7,a)--(9,a));
 +
}
 +
 +
fill((11,0)--(11,12)--(12,12)--(12,0)--cycle,lightgray);
 +
draw((12,0)--(12,12)--(11,12)--(11,0));
 +
draw((11,9)--(10,9)--(10,0));
 +
draw((11,11)--(12,11));
 +
draw((11,10)--(12,10));
 +
draw((11,9)--(12,9));
 +
for (int a = 1; a <= 8; ++a)
 +
{
 +
    draw((10,a)--(12,a));
 +
}
 +
 +
fill((14,0)--(14,10)--(15,10)--(15,0)--cycle,lightgray);
 +
draw((15,0)--(15,10)--(14,10)--(14,0));
 +
draw((14,8)--(13,8)--(13,0));
 +
draw((14,9)--(15,9));
 +
draw((14,8)--(15,8));
 +
for (int a = 1; a <= 7; ++a)
 +
{
 +
    draw((13,a)--(15,a));
 +
}
 +
 +
draw((16,0)--(0,0)--(0,13),black);
 +
label("Jan",(2,0),S);
 +
label("Feb",(5,0),S);
 +
label("Mar",(8,0),S);
 +
label("Apr",(11,0),S);
 +
label("May",(14,0),S);
 +
label("$\textbf{MONTHLY SALES}$",(8,14),N);
 +
label("S",(0,8),W);
 +
label("A",(0,7),W);
 +
label("L",(0,6),W);
 +
label("E",(0,5),W);
 +
label("S",(0,4),W);
 +
 +
draw((4,12.5)--(4,13.5)--(5,13.5)--(5,12.5)--cycle);
 +
label("Drums",(4,13),W);
 +
fill((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle,lightgray);
 +
draw((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle);
 +
label("Bugles",(15,13),W);
 +
</asy>
 +
 +
<math>\text{(A)}\ \text{Jan} \qquad \text{(B)}\ \text{Feb} \qquad \text{(C)}\ \text{Mar} \qquad \text{(D)}\ \text{Apr} \qquad \text{(E)}\ \text{May}</math>
  
 
[[1992 AJHSME Problems/Problem 21|Solution]]
 
[[1992 AJHSME Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== Problem 22 ==
 +
 +
Eight <math>1\times 1</math> square tiles are arranged as shown so their outside edges form a polygon with a perimeter of <math>14</math> units.  Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure.  Which of the following could be the perimeter of the new figure?
 +
 +
<asy>
 +
for (int a=1; a <= 4; ++a)
 +
{
 +
    draw((a,0)--(a,2));
 +
}
 +
draw((0,0)--(4,0));
 +
draw((0,1)--(5,1));
 +
draw((1,2)--(5,2));
 +
draw((0,0)--(0,1));
 +
draw((5,1)--(5,2));
 +
</asy>
 +
 +
<math>\text{(A)}\ 15 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20</math>
  
 
[[1992 AJHSME Problems/Problem 22|Solution]]
 
[[1992 AJHSME Problems/Problem 22|Solution]]
Line 173: Line 491:
  
 
== Problem 24 ==
 
== Problem 24 ==
 +
 +
Four circles of radius <math>3</math> are arranged as shown.  Their centers are the vertices of a square.  The area of the shaded region is closest to
 +
 +
<asy>
 +
fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray);
 +
fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white);
 +
fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white);
 +
fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white);
 +
fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white);
 +
 +
draw(circle((3,3),3));
 +
draw(circle((3,-3),3));
 +
draw(circle((-3,-3),3));
 +
draw(circle((-3,3),3));
 +
draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle);
 +
</asy>
 +
 +
<math>\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27</math>
  
 
[[1992 AJHSME Problems/Problem 24|Solution]]
 
[[1992 AJHSME Problems/Problem 24|Solution]]
Line 189: Line 525:
 
* [[AJHSME Problems and Solutions]]
 
* [[AJHSME Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
 +
 +
{{MAA Notice}}

Latest revision as of 12:36, 19 February 2020

1992 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

$\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=$

$\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10$

Solution

Problem 2

Which of the following is not equal to $\dfrac{5}{4}$?

$\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfrac{3}{12} \qquad \text{(D)}\ 1\dfrac{1}{5} \qquad \text{(E)}\ 1\dfrac{10}{40}$

Solution

Problem 3

What is the largest difference that can be formed by subtracting two numbers chosen from the set $\{ -16,-4,0,2,4,12 \}$?

$\text{(A)}\ 10 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 28 \qquad \text{(E)}\ 48$

Solution

Problem 4

During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were singles. What percent of her hits were singles?

$\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\%$

Solution

Problem 5

A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?

[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); [/asy]

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

Solution

Problem 6

Suppose that [asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$a$",(1,sqrt(3)-0.2),S); label("$b$",(sqrt(3)/10,0.1),ENE); label("$c$",(2-sqrt(3)/10,0.1),WNW); [/asy] means $a+b-c$. For example, [asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$5$",(1,sqrt(3)-0.2),S); label("$4$",(sqrt(3)/10,0.1),ENE); label("$6$",(2-sqrt(3)/10,0.1),WNW); [/asy] is $5+4-6 = 3$. Then the sum [asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$1$",(1,sqrt(3)-0.2),S); label("$3$",(sqrt(3)/10,0.1),ENE); label("$4$",(2-sqrt(3)/10,0.1),WNW); draw((3,0)--(5,0)--(4,sqrt(3))--cycle); label("$2$",(4,sqrt(3)-0.2),S); label("$5$",(3+sqrt(3)/10,0.1),ENE); label("$6$",(5-sqrt(3)/10,0.1),WNW); label("$+$",(2.5,-0.1),N); [/asy] is

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

Solution

Problem 7

The digit-sum of $998$ is $9+9+8=26$. How many 3-digit whole numbers, whose digit-sum is $26$, are even?

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

Solution

Problem 8

A store owner bought $1500$ pencils at $$0.10$ each. If he sells them for $$0.25$ each, how many of them must he sell to make a profit of exactly $$100.00$?

$\text{(A)}\ 400 \qquad \text{(B)}\ 667 \qquad \text{(C)}\ 1000 \qquad \text{(D)}\ 1500 \qquad \text{(E)}\ 1900$

Solution

Problem 9

The population of a small town is $480$. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town?

[asy] draw((0,13)--(0,0)--(20,0)); draw((3,0)--(3,10)--(8,10)--(8,0)); draw((3,5)--(8,5)); draw((11,0)--(11,5)--(16,5)--(16,0)); label("$\textbf{POPULATION}$",(10,11),N); label("$\textbf{F}$",(5.5,0),S); label("$\textbf{M}$",(13.5,0),S); [/asy]

$\text{(A)}\ 120 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 360$

Solution

Problem 10

An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown. The shaded area is

[asy] for (int a=0; a <= 3; ++a) { for (int b=0; b <= 3-a; ++b) { fill((a,b)--(a,b+1)--(a+1,b)--cycle,grey); } } for (int c=0; c <= 3; ++c) { draw((c,0)--(c,4-c),linewidth(1)); draw((0,c)--(4-c,c),linewidth(1)); draw((c+1,0)--(0,c+1),linewidth(1)); } label("$8$",(2,0),S); label("$8$",(0,2),W); [/asy]

$\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 64$

Solution

Problem 11

The bar graph shows the results of a survey on color preferences. What percent preferred blue?

[asy] for (int a = 1; a <= 6; ++a) { draw((-1.5,4*a)--(1.5,4*a)); } draw((0,28)--(0,0)--(32,0)); draw((3,0)--(3,20)--(6,20)--(6,0)); draw((9,0)--(9,24)--(12,24)--(12,0)); draw((15,0)--(15,16)--(18,16)--(18,0)); draw((21,0)--(21,24)--(24,24)--(24,0)); draw((27,0)--(27,16)--(30,16)--(30,0)); label("$20$",(-1.5,8),W); label("$40$",(-1.5,16),W); label("$60$",(-1.5,24),W); label("$\textbf{COLOR SURVEY}$",(16,26),N); label("$\textbf{F}$",(-6,25),W); label("$\textbf{r}$",(-6.75,22.4),W); label("$\textbf{e}$",(-6.75,19.8),W); label("$\textbf{q}$",(-6.75,17.2),W); label("$\textbf{u}$",(-6.75,15),W); label("$\textbf{e}$",(-6.75,12.4),W); label("$\textbf{n}$",(-6.75,9.8),W); label("$\textbf{c}$",(-6.75,7.2),W); label("$\textbf{y}$",(-6.75,4.6),W); label("D",(4.5,.2),N); label("E",(4.5,3),N); label("R",(4.5,5.8),N); label("E",(10.5,.2),N); label("U",(10.5,3),N); label("L",(10.5,5.8),N); label("B",(10.5,8.6),N); label("N",(16.5,.2),N); label("W",(16.5,3),N); label("O",(16.5,5.8),N); label("R",(16.5,8.6),N); label("B",(16.5,11.4),N); label("K",(22.5,.2),N); label("N",(22.5,3),N); label("I",(22.5,5.8),N); label("P",(22.5,8.6),N); label("N",(28.5,.2),N); label("E",(28.5,3),N); label("E",(28.5,5.8),N); label("R",(28.5,8.6),N); label("G",(28.5,11.4),N); [/asy]

$\text{(A)}\ 20\% \qquad \text{(B)}\ 24\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 36\% \qquad \text{(E)}\ 42\%$

Solution

Problem 12

The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used?

$\text{(A)}\ 6000 \qquad \text{(B)}\ 7500 \qquad \text{(C)}\ 24,000 \qquad \text{(D)}\ 30,000 \qquad \text{(E)}\ 37,500$

Solution

Problem 13

Five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$. The sum of the two lowest test scores is

$\text{(A)}\ 170 \qquad \text{(B)}\ 171 \qquad \text{(C)}\ 176 \qquad \text{(D)}\ 177 \qquad \text{(E)}\ \text{not determined by the information given}$

Solution

Problem 14

When four gallons are added to a tank that is one-third full, the tank is then one-half full. The capacity of the tank in gallons is

$\text{(A)}\ 8 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 48$

Solution

Problem 15

What is the $1992^\text{nd}$ letter in this sequence?

\[\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\cdots\]

$\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}$

Solution

Problem 16

[asy] draw(ellipse((0,-5),10,3)); fill((-10,-5)--(10,-5)--(10,5)--(-10,5)--cycle,white); draw(ellipse((0,0),10,3)); draw((10,0)--(10,-5)); draw((-10,0)--(-10,-5)); draw((0,0)--(7,-3*sqrt(51)/10)); label("10",(7/2,-3*sqrt(51)/20),NE); label("5",(-10,-3),E); [/asy]

Which cylinder has twice the volume of the cylinder shown above?

[asy] unitsize(4); draw(ellipse((0,-5),20,6)); fill((-20,-5)--(20,-5)--(20,5)--(-20,5)--cycle,white); draw(ellipse((0,0),20,6)); draw((20,0)--(20,-5)); draw((-20,0)--(-20,-5)); draw((0,0)--(14,-3*sqrt(51)/5)); label("20",(7,-3*sqrt(51)/10),NE); label("5",(-20,-4),E); label("(A)",(0,6),N); draw(ellipse((31,-7),10,3)); fill((21,-7)--(41,-7)--(41,7)--(21,7)--cycle,white); draw(ellipse((31,3),10,3)); draw((41,3)--(41,-7)); draw((21,3)--(21,-7)); draw((31,3)--(38,3-3*sqrt(51)/10)); label("10",(34.5,3-3*sqrt(51)/20),NE); label("10",(21,-4),E); label("(B)",(31,6),N); draw(ellipse((47,-15.5),5,3/2)); fill((42,-15.5)--(42,-15.5)--(42,15.5)--(42,15.5)--cycle,white); draw(ellipse((47,4.5),5,3/2)); draw((42,4.5)--(42,-15.5)); draw((52,4.5)--(52,-15.5)); draw((47,4.5)--(50.5,4.5-3*sqrt(51)/20)); label("5",(48.75,4.5-3*sqrt(51)/40),NE); label("10",(42,-6),E); label("(C)",(47,6),N); draw(ellipse((73,-10),20,6)); fill((53,-10)--(93,-10)--(93,5)--(53,5)--cycle,white); draw(ellipse((73,0),20,6)); draw((53,0)--(53,-10)); draw((93,0)--(93,-10)); draw((73,0)--(87,-3*sqrt(51)/5)); label("20",(80,-3*sqrt(51)/10),NE); label("10",(53,-6),E); label("(D)",(73,6),N); [/asy]

$\text{(E)}\ \text{None of the above}$

Solution

Problem 17

The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?

[asy] pair A,B,C; A=origin; B=(10,0); C=6.5*dir(15); dot(A); dot(B); dot(C); draw(B--A--C); draw(B--C,dashed); label("$6.5$",3.25*dir(15),NNW); label("$10$",(5,0),S); label("$s$",(8,1),NE); [/asy]

$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$

Solution

Problem 18

On a trip, a car traveled $80$ miles in an hour and a half, then was stopped in traffic for $30$ minutes, then traveled $100$ miles during the next $2$ hours. What was the car's average speed in miles per hour for the $4$-hour trip?

$\text{(A)}\ 45 \qquad \text{(B)}\ 50 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 90$

Solution

Problem 19

The distance between the $5^\text{th}$ and $26^\text{th}$ exits on an interstate highway is $118$ miles. If any two exits are at least $5$ miles apart, then what is the largest number of miles there can be between two consecutive exits that are between the $5^\text{th}$ and $26^\text{th}$ exits?

$\text{(A)}\ 8 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 47 \qquad \text{(E)}\ 98$

Solution

Problem 20

Which pattern of identical squares could NOT be folded along the lines shown to form a cube?

[asy] unitsize(12); draw((0,0)--(0,-1)--(1,-1)--(1,-2)--(2,-2)--(2,-3)--(4,-3)--(4,-2)--(3,-2)--(3,-1)--(2,-1)--(2,0)--cycle); draw((1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3)); draw((7,0)--(8,0)--(8,-1)--(11,-1)--(11,-2)--(8,-2)--(8,-3)--(7,-3)--cycle); draw((7,-1)--(8,-1)--(8,-2)--(7,-2)); draw((9,-1)--(9,-2)); draw((10,-1)--(10,-2)); draw((14,-1)--(15,-1)--(15,0)--(16,0)--(16,-1)--(18,-1)--(18,-2)--(17,-2)--(17,-3)--(16,-3)--(16,-2)--(14,-2)--cycle); draw((15,-2)--(15,-1)--(16,-1)--(16,-2)--(17,-2)--(17,-1)); draw((21,-1)--(22,-1)--(22,0)--(23,0)--(23,-2)--(25,-2)--(25,-3)--(22,-3)--(22,-2)--(21,-2)--cycle); draw((23,-1)--(22,-1)--(22,-2)--(23,-2)--(23,-3)); draw((24,-2)--(24,-3)); draw((28,-1)--(31,-1)--(31,0)--(32,0)--(32,-2)--(31,-2)--(31,-3)--(30,-3)--(30,-2)--(28,-2)--cycle); draw((32,-1)--(31,-1)--(31,-2)--(30,-2)--(30,-1)); draw((29,-1)--(29,-2)); label("(A)",(0,-0.5),W); label("(B)",(7,-0.5),W); label("(C)",(14,-0.5),W); label("(D)",(21,-0.5),W); label("(E)",(28,-0.5),W); [/asy]

Solution

Problem 21

Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?

[asy] unitsize(12); fill((2,0)--(2,9)--(3,9)--(3,0)--cycle,lightgray); draw((3,0)--(3,9)--(2,9)--(2,0)); draw((2,7)--(1,7)--(1,0)); draw((2,8)--(3,8)); draw((2,7)--(3,7)); for (int a = 1; a <= 6; ++a) { draw((1,a)--(3,a)); } fill((5,0)--(5,3)--(6,3)--(6,0)--cycle,lightgray); draw((6,0)--(6,3)--(5,3)--(5,0)); draw((5,3)--(5,5)--(4,5)--(4,0)); draw((4,4)--(5,4)); draw((4,3)--(5,3)); draw((4,2)--(6,2)); draw((4,1)--(6,1)); fill((8,0)--(8,6)--(9,6)--(9,0)--cycle,lightgray); draw((9,0)--(9,6)--(8,6)--(8,0)); draw((8,6)--(8,9)--(7,9)--(7,0)); draw((7,8)--(8,8)); draw((7,7)--(8,7)); draw((7,6)--(8,6)); for (int a = 1; a <= 5; ++a) { draw((7,a)--(9,a)); } fill((11,0)--(11,12)--(12,12)--(12,0)--cycle,lightgray); draw((12,0)--(12,12)--(11,12)--(11,0)); draw((11,9)--(10,9)--(10,0)); draw((11,11)--(12,11)); draw((11,10)--(12,10)); draw((11,9)--(12,9)); for (int a = 1; a <= 8; ++a) { draw((10,a)--(12,a)); } fill((14,0)--(14,10)--(15,10)--(15,0)--cycle,lightgray); draw((15,0)--(15,10)--(14,10)--(14,0)); draw((14,8)--(13,8)--(13,0)); draw((14,9)--(15,9)); draw((14,8)--(15,8)); for (int a = 1; a <= 7; ++a) { draw((13,a)--(15,a)); } draw((16,0)--(0,0)--(0,13),black); label("Jan",(2,0),S); label("Feb",(5,0),S); label("Mar",(8,0),S); label("Apr",(11,0),S); label("May",(14,0),S); label("$\textbf{MONTHLY SALES}$",(8,14),N); label("S",(0,8),W); label("A",(0,7),W); label("L",(0,6),W); label("E",(0,5),W); label("S",(0,4),W); draw((4,12.5)--(4,13.5)--(5,13.5)--(5,12.5)--cycle); label("Drums",(4,13),W); fill((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle,lightgray); draw((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle); label("Bugles",(15,13),W); [/asy]

$\text{(A)}\ \text{Jan} \qquad \text{(B)}\ \text{Feb} \qquad \text{(C)}\ \text{Mar} \qquad \text{(D)}\ \text{Apr} \qquad \text{(E)}\ \text{May}$

Solution

Problem 22

Eight $1\times 1$ square tiles are arranged as shown so their outside edges form a polygon with a perimeter of $14$ units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?

[asy] for (int a=1; a <= 4; ++a) { draw((a,0)--(a,2)); } draw((0,0)--(4,0)); draw((0,1)--(5,1)); draw((1,2)--(5,2)); draw((0,0)--(0,1)); draw((5,1)--(5,2)); [/asy]

$\text{(A)}\ 15 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$

Solution

Problem 23

If two dice are tossed, the probability that the product of the numbers showing on the tops of the dice is greater than $10$ is

$\text{(A)}\ \dfrac{3}{7} \qquad \text{(B)}\ \dfrac{17}{36} \qquad \text{(C)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{5}{8} \qquad \text{(E)}\ \dfrac{11}{12}$

Solution

Problem 24

Four circles of radius $3$ are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to

[asy] fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray); fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white); fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white); fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white); fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white); draw(circle((3,3),3)); draw(circle((3,-3),3)); draw(circle((-3,-3),3)); draw(circle((-3,3),3)); draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle); [/asy]

$\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27$

Solution

Problem 25

One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain?

$\text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 10$

Solution

See also

1992 AJHSME (Problems • Answer Key • Resources)
Preceded by
1991 AJHSME 		Followed by
1993 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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