Difference between revisions of "1990 AJHSME Problems"

Latest revision as of 14:03, 6 October 2021

1990 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

What is the smallest sum of two $3$ -digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?

$[asy] unitsize(12); draw((0,0)--(10,0)); draw((-1.5,1.5)--(-1.5,2.5)); draw((-1,2)--(-2,2)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle); draw((4,1)--(6,1)--(6,3)--(4,3)--cycle); draw((4,4)--(6,4)--(6,6)--(4,6)--cycle); draw((7,1)--(9,1)--(9,3)--(7,3)--cycle); draw((7,4)--(9,4)--(9,6)--(7,6)--cycle); [/asy]$

$\text{(A)}\ 947 \qquad \text{(B)}\ 1037 \qquad \text{(C)}\ 1047 \qquad \text{(D)}\ 1056 \qquad \text{(E)}\ 1245$

@@ Line 1: / Line 1: @@
+{{AJHSME Problems
+|year = 1990
+}}
 ==Problem 1==
+What is the smallest sum of two <math>3</math>-digit numbers that can be obtained by placing each of the six digits <math>4,5,6,7,8,9</math> in one of the six boxes in this addition problem?
+<asy>
+unitsize(12);
+draw((0,0)--(10,0)); draw((-1.5,1.5)--(-1.5,2.5)); draw((-1,2)--(-2,2));
+draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle);
+draw((4,1)--(6,1)--(6,3)--(4,3)--cycle); draw((4,4)--(6,4)--(6,6)--(4,6)--cycle);
+draw((7,1)--(9,1)--(9,3)--(7,3)--cycle); draw((7,4)--(9,4)--(9,6)--(7,6)--cycle);
+</asy>
+<math>\text{(A)}\ 947 \qquad \text{(B)}\ 1037 \qquad \text{(C)}\ 1047 \qquad \text{(D)}\ 1056 \qquad \text{(E)}\ 1245</math>
 [[1990 AJHSME Problems/Problem 1|Solution]]
@@ Line 12: / Line 27: @@
 == Problem 3 ==
+What fraction of the square is shaded?
+<asy>
+draw((0,0)--(0,3)--(3,3)--(3,0)--cycle);
+draw((0,2)--(2,2)--(2,0)); draw((0,1)--(1,1)--(1,0)); draw((0,0)--(3,3));
+fill((0,0)--(0,1)--(1,1)--cycle,grey);
+fill((1,0)--(1,1)--(2,2)--(2,0)--cycle,grey);
+fill((0,2)--(2,2)--(3,3)--(0,3)--cycle,grey);
+</asy>
+<math>\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{5} \qquad \text{(C)}\ \frac{5}{12} \qquad \text{(D)}\ \frac{3}{7} \qquad \text{(E)}\ \frac{1}{2}</math>
 [[1990 AJHSME Problems/Problem 3|Solution]]
 == Problem 4 ==
+Which of the following could '''not''' be the unit's digit <nowiki>[one's digit]</nowiki> of the square of a whole number?
+<math>\text{(A)}\ 1 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math>
 [[1990 AJHSME Problems/Problem 4|Solution]]
 == Problem 5 ==
+Which of the following is closest to the product <math>(.48017)(.48017)(.48017)</math>?
+<math>\text{(A)}\ 0.011 \qquad \text{(B)}\ 0.110 \qquad \text{(C)}\ 1.10 \qquad \text{(D)}\ 11.0 \qquad \text{(E)}\ 110</math>
 [[1990 AJHSME Problems/Problem 5|Solution]]
 == Problem 6 ==
+Which of these five numbers is the largest?
+<math>\text{(A)}\ 13579+\frac{1}{2468} \qquad \text{(B)}\ 13579-\frac{1}{2468} \qquad \text{(C)}\ 13579\times \frac{1}{2468}</math>
+<math>\text{(D)}\ 13579\div \frac{1}{2468} \qquad \text{(E)}\ 13579.2468</math>
 [[1990 AJHSME Problems/Problem 6|Solution]]
 == Problem 7 ==
+When three different numbers from the set <math>\{ -3, -2, -1, 4, 5 \} </math> are multiplied, the largest possible product is
+<math>\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 60</math>
 [[1990 AJHSME Problems/Problem 7|Solution]]
 == Problem 8 ==
+A dress originally priced at <math>80</math> dollars was put on sale for <math>25\% </math> off.  If <math>10\% </math> tax was added to the sale price, then the total selling price (in dollars) of the dress was
+<math>\text{(A)}\ \text{45 dollars} \qquad \text{(B)}\ \text{52 dollars} \qquad \text{(C)}\ \text{54 dollars} \qquad \text{(D)}\ \text{66 dollars} \qquad \text{(E)}\ \text{68 dollars}</math>
 [[1990 AJHSME Problems/Problem 8|Solution]]
 == Problem 9 ==
+The grading scale shown is used at Jones Junior High.  The fifteen scores in Mr. Freeman's class were: <cmath>\begin{tabular}[t]{lllllllll}
+, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75, \\
+, & 84, & 78, & 71, & 80, & 90. & & & \\
+\end{tabular}</cmath>
+In Mr. Freeman's class, what percent of the students received a grade of C?
+<cmath>\boxed{\begin{tabular}[t]{cc}
+A: & 93 - 100 \\
+B: & 85 - 92 \\
+C: & 75 - 84 \\
+D: & 70 - 74 \\
+F: & 0 - 69
+\end{tabular}}</cmath>
+<math>\text{(A)}\ 20\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 33\frac{1}{3}\% \qquad \text{(E)}\ 40\% </math>
 [[1990 AJHSME Problems/Problem 9|Solution]]
 == Problem 10 ==
+On this monthly calendar, the date behind one of the letters is added to the date behind <math>\text{C}</math>.  If this sum equals the sum of the dates behind <math>\text{A}</math> and <math>\text{B}</math>, then the letter is
+<asy>
+unitsize(12);
+draw((1,1)--(23,1));
+draw((0,5)--(23,5));
+draw((0,9)--(23,9));
+draw((0,13)--(23,13));
+for(int a=0; a<6; ++a)
+ {
+  draw((4a+2,0)--(4a+2,14));
+ }
+label("Tues.",(4,14),N); label("Wed.",(8,14),N); label("Thurs.",(12,14),N);
+label("Fri.",(16,14),N); label("Sat.",(20,14),N);
+label("C",(12,10.3),N); label("$\textbf{A}$",(16,10.3),N); label("Q",(12,6.3),N);
+label("S",(4,2.3),N); label("$\textbf{B}$",(8,2.3),N); label("P",(12,2.3),N);
+label("T",(16,2.3),N); label("R",(20,2.3),N);
+</asy>
+<math>\text{(A)}\ \text{P} \qquad \text{(B)}\ \text{Q} \qquad \text{(C)}\ \text{R} \qquad \text{(D)}\ \text{S} \qquad \text{(E)}\ \text{T}</math>
 [[1990 AJHSME Problems/Problem 10|Solution]]
 == Problem 11 ==
+The numbers on the faces of this cube are consecutive whole numbers.  The sums of the two numbers on each of the three pairs of opposite faces are equal.  The sum of the six numbers on this cube is
+<asy>
+draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
+draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
+draw((3,3)--(5,5));
+label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N);
+</asy>
+<math>\text{(A)}\ 75 \qquad \text{(B)}\ 76 \qquad \text{(C)}\ 78 \qquad \text{(D)}\ 80 \qquad \text{(E)}\ 81</math>
 [[1990 AJHSME Problems/Problem 11|Solution]]
 == Problem 12 ==
+There are twenty-four <math>4</math>-digit numbers that use each of the four digits <math>2</math>, <math>4</math>, <math>5</math>, and <math>7</math> exactly once.  Listed in numerical order from smallest to largest, the number in the <math>17\text{th}</math> position in the list is
+<math>\text{(A)}\ 4527 \qquad \text{(B)}\ 5724 \qquad \text{(C)}\ 5742 \qquad \text{(D)}\ 7245 \qquad \text{(E)}\ 7524</math>
 [[1990 AJHSME Problems/Problem 12|Solution]]
 == Problem 13 ==
+One proposal for new postage rates for a letter was <math>30</math> cents for the first ounce and <math>22</math> cents for each additional ounce (or fraction of an ounce).  The postage for a letter weighing <math>4.5</math> ounces was
+<math>\text{(A)}\ \text{96 cents} \qquad \text{(B)}\ \text{1.07 dollars} \qquad \text{(C)}\ \text{1.18 dollars} \qquad \text{(D)}\ \text{1.20 dollars} \qquad \text{(E)}\ \text{1.40 dollars}</math>
 [[1990 AJHSME Problems/Problem 13|Solution]]
 == Problem 14 ==
+A bag contains only blue balls and green balls.  There are <math>6</math> blue balls.  If the probability of drawing a blue ball at random from this bag is <math>\frac{1}{4}</math>, then the number of green balls in the bag is
+<math>\text{(A)}\ 12 \qquad \text{(B)}\ 18 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 36</math>
 [[1990 AJHSME Problems/Problem 14|Solution]]
 == Problem 15 ==
+The area of this figure is <math>100\text{ cm}^2</math>.  Its perimeter is
+<asy>
+draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1));
+draw((1,2)--(1,1)--(2,1)--(2,0),dashed);
+</asy>
+<center><nowiki>[figure consists of four identical squares]</nowiki></center>
+<math>\text{(A)}\ \text{20 cm} \qquad \text{(B)}\ \text{25 cm} \qquad \text{(C)}\ \text{30 cm} \qquad \text{(D)}\ \text{40 cm} \qquad \text{(E)}\ \text{50 cm}</math>
 [[1990 AJHSME Problems/Problem 15|Solution]]
 == Problem 16 ==
+<math>1990-1980+1970-1960+\cdots -20+10 =</math>
+<math>\text{(A)}\ -990 \qquad \text{(B)}\ -10 \qquad \text{(C)}\ 990 \qquad \text{(D)}\ 1000 \qquad \text{(E)}\ 1990</math>
 [[1990 AJHSME Problems/Problem 16|Solution]]
 == Problem 17 ==
+A straight concrete sidewalk is to be <math>3</math> feet wide, <math>60</math> feet long, and <math>3</math> '''inches''' thick.  How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
+<math>\text{(A)}\ 2 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ \text{more than 20}</math>
 [[1990 AJHSME Problems/Problem 17|Solution]]
 == Problem 18 ==
+Each corner of a rectangular prism is cut off.  Two (of the eight) cuts are shown.  How many edges does the new figure have?
+<asy>
+draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
+draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
+draw((3,3)--(5,5));
+draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1));
+draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1));
+</asy>
+<math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 42 \qquad \text{(E)}\ 48</math>
+''Assume that the planes cutting the prism do not intersect anywhere in or on the prism.''
 [[1990 AJHSME Problems/Problem 18|Solution]]
 == Problem 19 ==
+There are <math>120</math> seats in a row.  What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?
+<math>\text{(A)}\ 30 \qquad \text{(B)}\ 40 \qquad \text{(C)}\ 41 \qquad \text{(D)}\ 60 \qquad \text{(E)}\ 119</math>
 [[1990 AJHSME Problems/Problem 19|Solution]]
 == Problem 20 ==
+The annual incomes of <math>1,000</math> families range from <math>8200</math> dollars to <math>98,000</math> dollars.  In error, the largest income was entered on the computer as <math>980,000</math> dollars.  The difference between the mean of the incorrect data and the mean of the actual data is
+<math>\text{(A)}\ \text{882 dollars} \qquad \text{(B)}\ \text{980 dollars} \qquad \text{(C)}\ \text{1078 dollars} \qquad \text{(D)}\ \text{482,000 dollars} \qquad \text{(E)}\ \text{882,000 dollars}</math>
 [[1990 AJHSME Problems/Problem 20|Solution]]
 == Problem 21 ==
+A list of <math>8</math> numbers is formed by beginning with two given numbers.  Each new number in the list is the product of the two previous numbers.  Find the first number if the last three are shown:
+<cmath>\text{\underline{\hspace{3 mm}?\hspace{3 mm}}\hspace{1 mm},\hspace{1 mm} \underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm} \underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm} \underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm} \underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}16\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}64\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{1 mm}1024\hspace{1 mm}}}</cmath>
+<math>\text{(A)}\ \frac{1}{64} \qquad \text{(B)}\ \frac{1}{4} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 2 \qquad \text{(E)}\ 4</math>
 [[1990 AJHSME Problems/Problem 21|Solution]]
 == Problem 22 ==
+Several students are seated at a large circular table.  They pass around a bag containing <math>100</math> pieces of candy.  Each person receives the bag, takes one piece of candy and then passes the bag to the next person.  If Chris takes the first and last piece of candy, then the number of students at the table could be
+<math>\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 25</math>
 [[1990 AJHSME Problems/Problem 22|Solution]]
 == Problem 23 ==
+The graph relates the distance traveled <nowiki>[in miles]</nowiki> to the time elapsed <nowiki>[in hours]</nowiki> on a trip taken by an experimental airplane.  During which hour was the average speed of this airplane the largest?
+<asy>
+unitsize(12);
+for(int a=1; a<13; ++a)
+ {
+  draw((2a,-1)--(2a,1));
+ }
+draw((-1,4)--(1,4)); draw((-1,8)--(1,8)); draw((-1,12)--(1,12)); draw((-1,16)--(1,16));
+draw((0,0)--(0,17));
+draw((-5,0)--(33,0));
+label("$0$",(0,-1),S); label("$1$",(2,-1),S); label("$2$",(4,-1),S); label("$3$",(6,-1),S);
+label("$4$",(8,-1),S); label("$5$",(10,-1),S); label("$6$",(12,-1),S); label("$7$",(14,-1),S);
+label("$8$",(16,-1),S); label("$9$",(18,-1),S); label("$10$",(20,-1),S);
+label("$11$",(22,-1),S); label("$12$",(24,-1),S);
+label("Time in hours",(11,-2),S);
+label("$500$",(-1,4),W); label("$1000$",(-1,8),W); label("$1500$",(-1,12),W);
+label("$2000$",(-1,16),W);
+label(rotate(90)*"Distance traveled in miles",(-4,10),W);
+draw((0,0)--(2,3)--(4,7.2)--(6,8.5));
+draw((6,8.5)--(16,12.5)--(18,14)--(24,15));
+</asy>
+<math>\text{(A)}\ \text{first (0-1)} \qquad \text{(B)}\ \text{second (1-2)} \qquad \text{(C)}\ \text{third (2-3)} \qquad \text{(D)}\ \text{ninth (8-9)} \qquad \text{(E)}\ \text{last (11-12)}</math>
 [[1990 AJHSME Problems/Problem 23|Solution]]
 == Problem 24 ==
+Three <math>\Delta</math>'s and a <math>\diamondsuit </math> will balance nine <math>\bullet</math>'s.  One <math>\Delta </math> will balance a <math>\diamondsuit </math> and a <math>\bullet</math>.
+<asy>
+unitsize(5.5);
+fill((0,0)--(-4,-2)--(4,-2)--cycle,black);
+draw((-12,2)--(-12,0)--(12,0)--(12,2));
+draw(ellipse((-12,5),8,3)); draw(ellipse((12,5),8,3));
+label("$\Delta \hspace{2 mm}\Delta \hspace{2 mm}\Delta \hspace{2 mm}\diamondsuit $",(-12,6.5),S);
+label("$\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm} \bullet $",(12,5.2),N);
+label("$\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet \hspace{2 mm}\bullet $",(12,5.2),S);
+fill((44,0)--(40,-2)--(48,-2)--cycle,black);
+draw((34,2)--(34,0)--(54,0)--(54,2));
+draw(ellipse((34,5),6,3)); draw(ellipse((54,5),6,3));
+label("$\Delta $",(34,6.5),S);
+label("$\bullet \hspace{2 mm}\diamondsuit $",(54,6.5),S);
+</asy>
+How many <math>\bullet</math>'s will balance the two <math>\diamondsuit</math>'s in this balance?
+<asy>
+unitsize(5.5);
+fill((0,0)--(-4,-2)--(4,-2)--cycle,black);
+draw((-12,4)--(-12,2)--(12,-2)--(12,0));
+draw(ellipse((-12,7),6.5,3)); draw(ellipse((12,3),6.5,3));
+label("$?$",(-12,8.5),S);
+label("$\diamondsuit \hspace{2 mm}\diamondsuit $",(12,4.5),S);
+</asy>
+<math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5</math>
 [[1990 AJHSME Problems/Problem 24|Solution]]
 == Problem 25 ==
+How many different patterns can be made by shading exactly two of the nine squares?  Patterns that can be matched by flips and/or turns are not considered different.  For example, the patterns shown below are '''not''' considered different.
+<asy>
+fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray); fill((1,2)--(2,2)--(2,3)--(1,3)--cycle,gray);
+draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,linewidth(1));
+draw((2,0)--(2,3),linewidth(1)); draw((0,1)--(3,1),linewidth(1));
+draw((1,0)--(1,3),linewidth(1)); draw((0,2)--(3,2),linewidth(1));
+fill((6,0)--(8,0)--(8,1)--(6,1)--cycle,gray);
+draw((6,0)--(9,0)--(9,3)--(6,3)--cycle,linewidth(1));
+draw((8,0)--(8,3),linewidth(1)); draw((6,1)--(9,1),linewidth(1));
+draw((7,0)--(7,3),linewidth(1)); draw((6,2)--(9,2),linewidth(1));
+fill((14,1)--(15,1)--(15,3)--(14,3)--cycle,gray);
+draw((12,0)--(15,0)--(15,3)--(12,3)--cycle,linewidth(1));
+draw((14,0)--(14,3),linewidth(1)); draw((12,1)--(15,1),linewidth(1));
+draw((13,0)--(13,3),linewidth(1)); draw((12,2)--(15,2),linewidth(1));
+fill((18,1)--(19,1)--(19,3)--(18,3)--cycle,gray);
+draw((18,0)--(21,0)--(21,3)--(18,3)--cycle,linewidth(1));
+draw((20,0)--(20,3),linewidth(1)); draw((18,1)--(21,1),linewidth(1));
+draw((19,0)--(19,3),linewidth(1)); draw((18,2)--(21,2),linewidth(1));
+</asy>
+<math>\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 18</math>
 [[1990 AJHSME Problems/Problem 25|Solution]]
@@ Line 108: / Line 345: @@
 * [[AJHSME Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}

1990 AJHSME (Problems • Answer Key • Resources)
Preceded by 1989 AJHSME	Followed by 1991 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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Difference between revisions of "1990 AJHSME Problems"

Latest revision as of 14:03, 6 October 2021

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

See also