Difference between revisions of "1988 AJHSME Problems"
5849206328x (talk | contribs) m (→Problem 4: added choices) |
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+ | {{AJHSME Problems | ||
+ | |year = 1988 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
Line 43: | Line 46: | ||
<asy> | <asy> | ||
unitsize(12); | unitsize(12); | ||
+ | //Force a white background in middle even when transparent | ||
+ | fill((3,1)--(12,1)--(12,4)--(3,4)--cycle,white); | ||
+ | //Black Squares, Gray Border (blends better than white) | ||
for(int a=0; a<7; ++a) | for(int a=0; a<7; ++a) | ||
{ | { | ||
− | + | filldraw((2a,0)--(2a+1,0)--(2a+1,1)--(2a,1)--cycle,black,gray); | |
− | |||
} | } | ||
for(int b=7; b<15; ++b) | for(int b=7; b<15; ++b) | ||
{ | { | ||
− | + | filldraw((b,14-b)--(b+1,14-b)--(b+1,15-b)--(b,15-b)--cycle,black,gray); | |
} | } | ||
for(int c=1; c<7; ++c) | for(int c=1; c<7; ++c) | ||
{ | { | ||
− | + | filldraw((c,c)--(c+1,c)--(c+1,c+1)--(c,c+1)--cycle,black,gray); | |
} | } | ||
− | for(int | + | filldraw((6,4)--(7,4)--(7,5)--(6,5)--cycle,black,gray); |
+ | filldraw((7,5)--(8,5)--(8,6)--(7,6)--cycle,black,gray); | ||
+ | filldraw((8,4)--(9,4)--(9,5)--(8,5)--cycle,black,gray); | ||
+ | //White Squares, Black Border | ||
+ | filldraw((7,4)--(8,4)--(8,5)--(7,5)--cycle,white,black); | ||
+ | for(int a=0; a<7; ++a) | ||
+ | { | ||
+ | filldraw((2a+1,0)--(2a+2,0)--(2a+2,1)--(2a+1,1)--cycle,white,black); | ||
+ | } | ||
+ | for(int b=9; b<15; ++b) | ||
+ | { | ||
+ | filldraw((b-1,14-b)--(b,14-b)--(b,15-b)--(b-1,15-b)--cycle,white,black); | ||
+ | } | ||
+ | for(int c=1; c<7; ++c) | ||
{ | { | ||
− | + | filldraw((c+1,c)--(c+2,c)--(c+2,c+1)--(c+1,c+1)--cycle,white,black); | |
} | } | ||
− | |||
− | |||
− | |||
label("same",(6.3,2.45),N); | label("same",(6.3,2.45),N); | ||
label("pattern here",(7.5,1.4),N); | label("pattern here",(7.5,1.4),N); | ||
Line 120: | Line 135: | ||
== Problem 9 == | == Problem 9 == | ||
+ | |||
+ | An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles? | ||
+ | |||
+ | <asy> | ||
+ | for(int a=0; a<12; ++a) | ||
+ | { | ||
+ | draw((a,0)--(a,6)); | ||
+ | } | ||
+ | for(int b=0; b<7; ++b) | ||
+ | { | ||
+ | draw((0,b)--(11,b)); | ||
+ | } | ||
+ | draw((0,6)--(2,6)--(1,4)--cycle,linewidth(1)); | ||
+ | draw((3,4)--(3,6)--(5,4)--cycle,linewidth(1)); | ||
+ | draw((0,1)--(3,2)--(6,1)--cycle,linewidth(1)); | ||
+ | draw((7,4)--(6,6)--(9,4)--cycle,linewidth(1)); | ||
+ | draw((8,1)--(9,3)--(10,0)--cycle,linewidth(1)); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5</math> | ||
[[1988 AJHSME Problems/Problem 9|Solution]] | [[1988 AJHSME Problems/Problem 9|Solution]] | ||
Line 125: | Line 160: | ||
== Problem 10 == | == Problem 10 == | ||
− | Chris' birthday is on a Thursday this year. What day of the week will it be <math>60</math> days after her birthday? | + | Chris's birthday is on a Thursday this year. What day of the week will it be <math>60</math> days after her birthday? |
<math>\text{(A)}\ \text{Monday} \qquad \text{(B)}\ \text{Wednesday} \qquad \text{(C)}\ \text{Thursday} \qquad \text{(D)}\ \text{Friday} \qquad \text{(E)}\ \text{Saturday}</math> | <math>\text{(A)}\ \text{Monday} \qquad \text{(B)}\ \text{Wednesday} \qquad \text{(C)}\ \text{Thursday} \qquad \text{(D)}\ \text{Friday} \qquad \text{(E)}\ \text{Saturday}</math> | ||
Line 143: | Line 178: | ||
Suppose the estimated <math>20</math> billion dollar cost to send a person to the planet Mars is shared equally by the <math>250</math> million people in the U.S. Then each person's share is | Suppose the estimated <math>20</math> billion dollar cost to send a person to the planet Mars is shared equally by the <math>250</math> million people in the U.S. Then each person's share is | ||
− | <math>\text{(A)}\ | + | <math>\text{(A)}\ 20\text{ dollars} \qquad \text{(B)}\ 50\text{ dollars} \qquad \text{(C)}\ 80\text{ dollars} \qquad \text{(D)}\ 100\text{ dollars} \qquad \text{(E)}\ 125\text{ dollars}</math> |
[[1988 AJHSME Problems/Problem 12|Solution]] | [[1988 AJHSME Problems/Problem 12|Solution]] | ||
Line 191: | Line 226: | ||
== Problem 17 == | == Problem 17 == | ||
+ | |||
+ | The shaded region formed by the two intersecting perpendicular rectangles, in square units, is | ||
+ | |||
+ | <asy> | ||
+ | fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4.5)--(6,4.5)--(6,2)--(0,2)--cycle,black); | ||
+ | label("2",(0,.9),W); | ||
+ | label("3",(7.3,4.5),N); | ||
+ | draw((0,-3.3)--(0,-5.3),linewidth(1)); | ||
+ | draw((0,-4.3)--(3.7,-4.3),linewidth(1)); | ||
+ | label("10",(4.7,-3.7),S); | ||
+ | draw((5.7,-4.3)--(10,-4.3),linewidth(1)); | ||
+ | draw((10,-3.3)--(10,-5.3),linewidth(1)); | ||
+ | draw((11,4.5)--(13,4.5),linewidth(1)); | ||
+ | draw((12,4.5)--(12,2),linewidth(1)); | ||
+ | label("8",(11.3,1),E); | ||
+ | draw((12,0)--(12,-3.5),linewidth(1)); | ||
+ | draw((11,-3.5)--(13,-3.5),linewidth(1)); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 23 \qquad \text{(B)}\ 38 \qquad \text{(C)}\ 44 \qquad \text{(D)}\ 46 \qquad \text{(E)}\ \text{unable to be determined from the information given}</math> | ||
[[1988 AJHSME Problems/Problem 17|Solution]] | [[1988 AJHSME Problems/Problem 17|Solution]] | ||
Line 211: | Line 266: | ||
== Problem 20 == | == Problem 20 == | ||
+ | |||
+ | The glass gauge on a cylindrical coffee maker shows that there are <math>45</math> cups left when the coffee maker is <math>36\% </math> full. How many cups of coffee does it hold when it is full? | ||
+ | |||
+ | <math>\text{(A)}\ 80 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 125 \qquad \text{(D)}\ 130 \qquad \text{(E)}\ 262</math> | ||
+ | |||
+ | <asy> | ||
+ | draw((5,0)..(0,-1.3)..(-5,0)); | ||
+ | draw((5,0)--(5,10)); draw((-5,0)--(-5,10)); | ||
+ | draw(ellipse((0,10),5,1.3)); | ||
+ | draw(circle((.3,1.3),.4)); | ||
+ | draw((-.1,1.7)--(-.1,7.9)--(.7,7.9)--(.7,1.7)--cycle); | ||
+ | fill((-.1,1.7)--(-.1,4)--(.7,4)--(.7,1.7)--cycle,black); | ||
+ | draw((-2,11.3)--(2,11.3)..(2.6,11.9)..(2,12.2)--(-2,12.2)..(-2.6,11.9)..cycle); | ||
+ | </asy> | ||
[[1988 AJHSME Problems/Problem 20|Solution]] | [[1988 AJHSME Problems/Problem 20|Solution]] | ||
Line 234: | Line 303: | ||
<math>\text{(D)}\ \text{The sale price is lower than the original price.}</math> | <math>\text{(D)}\ \text{The sale price is lower than the original price.}</math> | ||
− | <math>\text{(E)}\ \text{The | + | <math>\text{(E)}\ \text{The sale price is the same as the original price.}</math> |
[[1988 AJHSME Problems/Problem 22|Solution]] | [[1988 AJHSME Problems/Problem 22|Solution]] | ||
Line 240: | Line 309: | ||
== Problem 23 == | == Problem 23 == | ||
− | Maria buys computer disks at a price of <math>4</math> for < | + | Maria buys computer disks at a price of <math>4</math> for <math><dollar>5</math> and sells them at a price of <math>3</math> for <math><dollar></math><math>5</math>. How many computer disks must she sell in order to make a profit of <math><dollar>100</math>? |
<math>\text{(A)}\ 100 \qquad \text{(B)}\ 120 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 1200</math> | <math>\text{(A)}\ 100 \qquad \text{(B)}\ 120 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 1200</math> | ||
Line 247: | Line 316: | ||
== Problem 24 == | == Problem 24 == | ||
+ | |||
+ | <asy> | ||
+ | unitsize(15); | ||
+ | for (int a=0; a<6; ++a) | ||
+ | { | ||
+ | draw(2*dir(60a)--2*dir(60a+60),linewidth(1)); | ||
+ | } | ||
+ | draw((1,1.7320508075688772935274463415059)--(1,3.7320508075688772935274463415059)--(-1,3.7320508075688772935274463415059)--(-1,1.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | fill((.4,1.7320508075688772935274463415059)--(0,3.35)--(-.4,1.7320508075688772935274463415059)--cycle,black); | ||
+ | label("1.",(0,-2),S); | ||
+ | draw(arc((1,1.7320508075688772935274463415059),1,90,300,CW)); | ||
+ | draw((1.5,0.86602540378443864676372317075294)--(1.75,1.7)); | ||
+ | draw((1.5,0.86602540378443864676372317075294)--(2.2,1)); | ||
+ | draw((7,0)--(6,1.7320508075688772935274463415059)--(4,1.7320508075688772935274463415059)--(3,0)--(4,-1.7320508075688772935274463415059)--(6,-1.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | draw((7,0)--(6,1.7320508075688772935274463415059)--(7.7320508075688772935274463415059,2.7320508075688772935274463415059)--(8.7320508075688772935274463415059,1)--cycle,linewidth(1)); | ||
+ | label("2.",(5,-2),S); | ||
+ | draw(arc((7,0),1,30,240,CW)); | ||
+ | draw((6.5,-0.86602540378443864676372317075294)--(7.1,-.7)); | ||
+ | draw((6.5,-0.86602540378443864676372317075294)--(6.8,-1.5)); | ||
+ | draw((14,0)--(13,1.7320508075688772935274463415059)--(11,1.7320508075688772935274463415059)--(10,0)--(11,-1.7320508075688772935274463415059)--(13,-1.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | draw((14,0)--(13,-1.7320508075688772935274463415059)--(14.7320508075688772935274463415059,-2.7320508075688772935274463415059)--(15.7320508075688772935274463415059,-1)--cycle,linewidth(1)); | ||
+ | label("3.",(12,-2.5),S); | ||
+ | draw((21,0)--(20,1.7320508075688772935274463415059)--(18,1.7320508075688772935274463415059)--(17,0)--(18,-1.7320508075688772935274463415059)--(20,-1.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | draw((18,-1.7320508075688772935274463415059)--(20,-1.7320508075688772935274463415059)--(20,-3.7320508075688772935274463415059)--(18,-3.7320508075688772935274463415059)--cycle,linewidth(1)); | ||
+ | label("4.",(19,-4),S); | ||
+ | </asy> | ||
+ | |||
+ | The square in the first diagram "rolls" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram <math>4</math>? | ||
+ | |||
+ | <asy> | ||
+ | unitsize(12); | ||
+ | label("(A)",(0,0),W); | ||
+ | fill((1,-1)--(1,1)--(5,0)--cycle,black); | ||
+ | label("(B)",(6,0),E); | ||
+ | fill((9,-2)--(11,-2)--(10,1)--cycle,black); | ||
+ | label("(C)",(14,0),E); | ||
+ | fill((17,1)--(19,1)--(18,-1.8)--cycle,black); | ||
+ | label("(D)",(22,0),E); | ||
+ | fill((25,-1)--(27,-2)--(28,1)--cycle,black); | ||
+ | label("(E)",(31,0),E); | ||
+ | fill((33,0)--(37,1)--(37,-1)--cycle,black); | ||
+ | </asy> | ||
[[1988 AJHSME Problems/Problem 24|Solution]] | [[1988 AJHSME Problems/Problem 24|Solution]] | ||
Line 259: | Line 370: | ||
== See also == | == See also == | ||
+ | {{AJHSME box|year=1988|before=[[1987 AJHSME Problems|1987 AJHSME]]|after=[[1989 AJHSME Problems|1989 AJHSME]]}} | ||
* [[AJHSME]] | * [[AJHSME]] | ||
* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
− | |||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | |||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 15:23, 17 January 2023
1988 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
The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of
Problem 2
The product
Problem 3
Problem 4
The figure consists of alternating light and dark squares. The number of dark squares exceeds the number of light squares by
Problem 5
If is a right angle, then this protractor indicates that the measure of
is approximately
Problem 6
Problem 7
is closest to
Problem 8
Betty used a calculator to find the product . She forgot to enter the decimal points. The calculator showed
. If Betty had entered the decimal points correctly, the answer would have been
Problem 9
An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?
Problem 10
Chris's birthday is on a Thursday this year. What day of the week will it be days after her birthday?
Problem 11
is
Problem 12
Suppose the estimated billion dollar cost to send a person to the planet Mars is shared equally by the
million people in the U.S. Then each person's share is
Problem 13
If rose bushes are spaced about foot apart, approximately how many bushes are needed to surround a circular patio whose radius is
feet?
Problem 14
and
are whole numbers and
. The largest possible value of
is
Problem 15
The reciprocal of is
Problem 16
Placing no more than one in each small square, what is the greatest number of
's that can be put on the grid shown without getting three
's in a row vertically, horizontally, or diagonally?
Problem 17
The shaded region formed by the two intersecting perpendicular rectangles, in square units, is
Problem 18
The average weight of boys is
pounds and the average weight of
girls is
pounds. The average weight of the
children is
Problem 19
What is the number in the arithmetic sequence:
?
Problem 20
The glass gauge on a cylindrical coffee maker shows that there are cups left when the coffee maker is
full. How many cups of coffee does it hold when it is full?
Problem 21
A fifth number, , is added to the set
to make the mean of the set of five numbers equal to its median. The number of possible values of
is
Problem 22
Tom's Hat Shoppe increased all original prices by . Now the shoppe is having a sale where all prices are
off these increased prices. Which statement best describes the sale price of an item?
Problem 23
Maria buys computer disks at a price of for
and sells them at a price of
for
. How many computer disks must she sell in order to make a profit of
?
Problem 24
The square in the first diagram "rolls" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram ?
Problem 25
A palindrome is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: ,
, and
. How many times during a
-hour period will be palindromes?
See also
1988 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by 1987 AJHSME |
Followed by 1989 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.