Difference between revisions of "1987 AJHSME Problems"
5849206328x (talk | contribs) |
(→Problem 3) |
||
(20 intermediate revisions by 9 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AJHSME Problems | ||
+ | |year = 1987 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
Line 54: | Line 57: | ||
== Problem 7 == | == Problem 7 == | ||
+ | |||
+ | The large cube shown is made up of <math>27</math> identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is | ||
+ | |||
+ | <math>\text{(A)}\ 10 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 22 \qquad \text{(E)}\ 24</math> | ||
+ | |||
+ | <asy> | ||
+ | unitsize(36); | ||
+ | draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); | ||
+ | draw((3,0)--(5.2,1.4)--(5.2,4.4)--(3,3)); | ||
+ | draw((0,3)--(2.2,4.4)--(5.2,4.4)); | ||
+ | fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black); | ||
+ | fill((0,2)--(0,3)--(1,3)--(1,2)--cycle,black); | ||
+ | fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black); | ||
+ | fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); | ||
+ | fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black); | ||
+ | draw((1,3)--(3.2,4.4)); | ||
+ | draw((2,3)--(4.2,4.4)); | ||
+ | draw((.733333333,3.4666666666)--(3.73333333333,3.466666666666)); | ||
+ | draw((1.466666666,3.9333333333)--(4.466666666,3.9333333333)); | ||
+ | fill((1.73333333,3.46666666666)--(2.7333333333,3.46666666666)--(3.46666666666,3.93333333333)--(2.46666666666,3.93333333333)--cycle,black); | ||
+ | fill((3,1)--(3.733333333333,1.466666666666)--(3.73333333333,2.46666666666)--(3,2)--cycle,black); | ||
+ | fill((3.73333333333,.466666666666)--(4.466666666666,.93333333333)--(4.46666666666,1.93333333333)--(3.733333333333,1.46666666666)--cycle,black); | ||
+ | fill((3.73333333333,2.466666666666)--(4.466666666666,2.93333333333)--(4.46666666666,3.93333333333)--(3.733333333333,3.46666666666)--cycle,black); | ||
+ | fill((4.466666666666,1.9333333333333)--(5.2,2.4)--(5.2,3.4)--(4.4666666666666,2.9333333333333)--cycle,black); | ||
+ | </asy> | ||
[[1987 AJHSME Problems/Problem 7|Solution]] | [[1987 AJHSME Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | |||
+ | If <math>\text{A}</math> and <math>\text{B}</math> are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is | ||
+ | |||
+ | <cmath>\begin{tabular}[t]{cccc} | ||
+ | 9 & 8 & 7 & 6 \\ | ||
+ | & A & 3 & 2 \\ | ||
+ | & & B & 1 \\ \hline | ||
+ | \end{tabular}</cmath> | ||
+ | |||
+ | |||
+ | <math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{depends on the values of A and B}</math> | ||
[[1987 AJHSME Problems/Problem 8|Solution]] | [[1987 AJHSME Problems/Problem 8|Solution]] | ||
Line 126: | Line 165: | ||
== Problem 15 == | == Problem 15 == | ||
− | The sale ad read: "Buy three tires at the regular price and get the fourth tire for | + | The sale ad read: "Buy three tires at the regular price and get the fourth tire for three dollars;." Sam paid <math>240\text{ dollars}</math> for a set of four tires at the sale. What was the regular price of one tire? |
<math>\text{(A)}\ 59.25\text{ dollars} \qquad \text{(B)}\ 60\text{ dollars} \qquad \text{(C)}\ 70\text{ dollars} \qquad \text{(D)}\ 79\text{ dollars} \qquad \text{(E)}\ 80\text{ dollars}</math> | <math>\text{(A)}\ 59.25\text{ dollars} \qquad \text{(B)}\ 60\text{ dollars} \qquad \text{(C)}\ 70\text{ dollars} \qquad \text{(D)}\ 79\text{ dollars} \qquad \text{(E)}\ 80\text{ dollars}</math> | ||
Line 149: | Line 188: | ||
However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2? | However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2? | ||
− | <math>\text{(A)}\ | + | <math>\text{(A)}\ \text{Abby} \qquad \text{(B)}\ \text{Bret} \qquad \text{(C)}\ \text{Carl} \qquad \text{(D)}\ \text{Dana} \qquad \text{(E)}\ \text{There is not enough information to be sure.}</math> |
[[1987 AJHSME Problems/Problem 17|Solution]] | [[1987 AJHSME Problems/Problem 17|Solution]] | ||
Line 155: | Line 194: | ||
== Problem 18 == | == Problem 18 == | ||
− | Half the people in a room left. One third of those remaining started to dance. There were then <math>12</math> people who were not dancing. The original number of people in the room was | + | Half the people in a room left. One third of those remaining started to dance. There were then <math>12</math> people who were not dancing. The original number of people in the room was what? |
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 42 \qquad \text{(E)}\ 72</math> | <math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 42 \qquad \text{(E)}\ 72</math> | ||
Line 201: | Line 240: | ||
fill(B..(4,3)..(5,0)--C--cycle,black); | fill(B..(4,3)..(5,0)--C--cycle,black); | ||
draw((0,5)--D--(5,0)); | draw((0,5)--D--(5,0)); | ||
+ | label("A",A,NW); | ||
+ | label("B",B,NE); | ||
+ | label("C",C,S); | ||
+ | label("D",D,SW); | ||
</asy> | </asy> | ||
Line 208: | Line 251: | ||
== Problem 23 == | == Problem 23 == | ||
+ | |||
+ | Assume the adjoining chart shows the <math>1980</math> U.S. population, in millions, for each region by ethnic group. To the nearest percent, what percent of the U.S. Black population lived in the South? | ||
+ | |||
+ | <cmath>\begin{tabular}[t]{c|cccc} | ||
+ | & NE & MW & South & West \\ \hline | ||
+ | White & 42 & 52 & 57 & 35 \\ | ||
+ | Black & 5 & 5 & 15 & 2 \\ | ||
+ | Asian & 1 & 1 & 1 & 3 \\ | ||
+ | Other & 1 & 1 & 2 & 4 | ||
+ | \end{tabular}</cmath> | ||
+ | |||
+ | <math>\text{(A)}\ 20\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 56\% \qquad \text{(E)}\ 80\% </math> | ||
[[1987 AJHSME Problems/Problem 23|Solution]] | [[1987 AJHSME Problems/Problem 23|Solution]] | ||
Line 227: | Line 282: | ||
[[1987 AJHSME Problems/Problem 25|Solution]] | [[1987 AJHSME Problems/Problem 25|Solution]] | ||
− | == See | + | == See Also == |
+ | {{AJHSME box|year=1987|before=[[1986 AJHSME Problems|1986 AJHSME]]|after=[[1988 AJHSME Problems|1988 AJHSME]]}} | ||
* [[AJHSME]] | * [[AJHSME]] | ||
* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
− | |||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | |||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 09:35, 31 October 2021
1987 AJHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See Also
Problem 1
Problem 2
Problem 3
Problem 4
Martians measure angles in clerts. There are clerts in a full circle. How many clerts are there in a right angle?
Problem 5
The area of the rectangular region is
Problem 6
The smallest product one could obtain by multiplying two numbers in the set is
Problem 7
The large cube shown is made up of identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is
Problem 8
If and are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is
Problem 9
When finding the sum , the least common denominator used is
Problem 10
Problem 11
The sum is between
Problem 12
What fraction of the large by rectangular region is shaded?
Problem 13
Which of the following fractions has the largest value?
Problem 14
A computer can do additions per second. How many additions can it do in one hour?
Problem 15
The sale ad read: "Buy three tires at the regular price and get the fourth tire for three dollars;." Sam paid for a set of four tires at the sale. What was the regular price of one tire?
Problem 16
Joyce made of her first shots in the first three games of this basketball game, so her seasonal shooting average was . In her next game, she took shots and raised her seasonal shooting average to . How many of these shots did she make?
Problem 17
Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:
"Bret is next to Carl." "Abby is between Bret and Carl."
However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?
Problem 18
Half the people in a room left. One third of those remaining started to dance. There were then people who were not dancing. The original number of people in the room was what?
Problem 19
A calculator has a squaring key which replaces the current number displayed with its square. For example, if the display is and the key is depressed, then the display becomes . If the display reads , how many times must you depress the key to produce a displayed number greater than ?
Problem 20
"If a whole number is not prime, then the whole number is not prime." A value of which shows this statement to be false is
Problem 21
Suppose means , the reciprocal of . For example, . How many of the following statements are true?
i) ii) iii) iv)
Problem 22
is a rectangle, is the center of the circle, and is on the circle. If and , then the area of the shaded region is between
Problem 23
Assume the adjoining chart shows the U.S. population, in millions, for each region by ethnic group. To the nearest percent, what percent of the U.S. Black population lived in the South?
Problem 24
A multiple choice examination consists of questions. The scoring is for each correct answer, for each incorrect answer, and for each unanswered question. John's score on the examination is . What is the maximum number of questions he could have answered correctly?
Problem 25
Ten balls numbered to are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is
See Also
1987 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by 1986 AJHSME |
Followed by 1988 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.