1988 AMC 8

Part of AMC 8

Subcontests

(25)

1

1988 AJHSME #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:

\boxed{1:01}

\boxed{12:21}

. How many times during a 12-hour period will be palindromes?

\text{(A)}\ 57\qquad\text{(B)}\ 60\qquad\text{(C)}\ 63\qquad\text{(D)}\ 90\qquad\text{(E)}\ 93

1

1988 AJHSME #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

4

?
[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]

1

1988 AJHSME #23

Maria buys computer disks at a price of 4 for 5 dollars and sells them at a price of 3 for 5 dollars. How many computer disks must she sell in order to make a profit of 100 dolars?

\text{(A)}\ 100\qquad\text{(B)}\ 120\qquad\text{(C)}\ 200\qquad\text{(D)}\ 240\qquad\text{(E)}\ 1200

1

1988 AJHSME #22

Tom's Hat Shoppe increased all original prices by

25\%

. Now the shoppe is having a sale where all prices are

20\%

off these increased prices. Which statement best describes the sale price of an item?

\text{(A)}\ \text{The sale price is }5\%\text{ higher than the original price.}

\text{(B)}\ \text{The sale price is higher than the original price, but by less than }5\% .

\text{(C)}\ \text{The sale price is higher than the original price, but by more than }5\% .

\text{(D)}\ \text{The sale price is lower than the original price.}

\text{(E)}\ \text{The sale price is the same as the original price.}

1

1988 AJHSME #21

A fifth number,

n

, is added to the set

\{ 3,6,9,10\}

to make the mean of the set of five numbers equal to its median. The number of possible values of

n

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

1

1988 AJHSME #20

The glass gauge on a cylindrical coffee maker shows that there are 45 cups left when the coffee maker is

36\%

full. How many cups of coffee does it hold when it is full?
[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]

\text{(A)}\ 80\qquad\text{(B)}\ 100\qquad\text{(C)}\ 125\qquad\text{(D)}\ 130\qquad\text{(E)}\ 262

1

1988 AJHSME #19

100th

number in the arithmetic sequence:

1,5,9,13,17,21,25,...

\text{(A)}\ 397\qquad\text{(B)}\ 399\qquad\text{(C)}\ 401\qquad\text{(D)}\ 403\qquad\text{(E)}\ 405

1

1988 AJHSME #18

The average weight of 6 boys is 150 pounds and the average weight of 4 girls is 120 pounds. The average weight of the 10 children is

\text{(A)}\ 135\text{ pounds}\qquad\text{(B)}\ 137\text{ pounds}\qquad\text{(C)}\ 138\text{ pounds}\qquad\text{(D)}\ 140\text{ pounds}\qquad\text{(E)}\ 141\text{ pounds}

1

1988 AJHSME #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]

\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}

1

1988 AJHSME #16

[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b)); }[/asy]
Placing no more than one

x

in each small square, what is the greatest number of

x

's that can be put on the grid shown without getting three

x

's in a row vertically, horizontally, or diagonally?

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

1

1988 AJHSME #15

The reciprocal of

\left(\frac{1}{2}+\frac{1}{3}\right)

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

1

1988 AJHSME #14

\diamondsuit

\Delta

are whole numbers and

\diamondsuit\times\Delta =36

. The largest possible value of

\diamondsuit+\Delta

\text{(A)}\ 12\qquad\text{(B)}\ 13\qquad\text{(C)}\ 15\qquad\text{(D)}\ 20\ \qquad\text{(E)}\ 37

1

1988 AJHSME #13

If rose bushes are spaced about

1

foot apart, approximately how many bushes are needed to surround a circular patio whose radius is

12

\text{(A)}\ 12\qquad\text{(B)}\ 38\qquad\text{(C)}\ 48\qquad\text{(D)}\ 75\qquad\text{(E)}\ 450

1

1988 AJHSME #12

Suppose the estimated

20

billion dollar cost to send a person to the planet Mars is shared equally by the

250

million people in the U.S. Then each person's share is

\text{(A)}\ 40\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}

1

1988 AJHSME #11

\sqrt{164}

\text{(A)}\ 42\qquad\text{(B)}\ \text{less than }10\qquad\text{(C)}\ \text{between }10\text{ and }11\qquad\text{(D)}\ \text{between }11\text{ and }12\qquad\text{(E)}\ \text{between }12\text{ and }13

1

1988 AJHSME #10

Chris' birthday is on a Thursday this year. What day of the week will it be

60

days after her birthday?

\text{(A)}\ \text{Monday}\qquad\text{(B)}\ \text{Wednesday}\qquad\text{(C)}\ \text{Thursday}\qquad\text{(D)}\ \text{Friday}\qquad\text{(E)}\ \text{Saturday}

1

1988 AJHSME #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]

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

1

1988 AJHSME #8

Betty used a calculator to find the product

0.075\times 2.56

. She forgot to enter the decimal points. The calculator showed

19200

. If Betty had entered the decimal points correctly, the answer would have been

\text{(A)}\ .0192\qquad\text{(B)}\ .192\qquad\text{(C)}\ 1.92\qquad\text{(D)}\ 19.2\qquad\text{(E)}\ 192

1

1988 AJHSME #7

2.46\times 8.163\times (5.17+4.829)

\text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500

1

1988 AJHSME #6

\frac{(.2)^{3}}{(.02)^{2}}=

\text{(A)}\ .2\qquad\text{(B)}\ 2\qquad\text{(C)}\ 10\qquad\text{(D)}\ 15\qquad\text{(E)}\ 20

1

1988 AJHSME #5

\angle\text{CBD}

is a right angle, then this protractor indicates that the measure of

\angle\text{ABC}

is approximately
[asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label("O",(-2.5,0),W); label("A",A,W); label("B",B,S); label("C",C,W); label("D",D,E); label("0",(-1.8,0),W); label("20",(-1.7,.5),NW); label("160",(1.6,.5),NE); label("180",(1.7,0),E);[/asy]

\text{(A)}\ 20^\circ\qquad\text{(B)}\ 40^\circ\qquad\text{(C)}\ 50^\circ\qquad\text{(D)}\ 70^\circ\qquad\text{(E)}\ 120^\circ

1

1988 AJHSME #4

The figure consists of alternating light and dark squares. The number of dark squares exceeds the number of light squares by

\text{(A)}\ 7\qquad\text{(B)}\ 8\qquad\text{(C)}\ 9\qquad\text{(D)}\ 10\qquad\text{(E)}\ 11

[asy] unitsize(12); for(int a=0; a<7; ++a) { fill((2a,0)--(2a+1,0)--(2a+1,1)--(2a,1)--cycle,black); draw((2a+1,0)--(2a+2,0)); } for(int b=7; b<15; ++b) { fill((b,14-b)--(b+1,14-b)--(b+1,15-b)--(b,15-b)--cycle,black); } for(int c=1; c<7; ++c) { fill((c,c)--(c+1,c)--(c+1,c+1)--(c,c+1)--cycle,black); } for(int d=1; d<6; ++d) { draw((2d+1,1)--(2d+2,1)); } fill((6,4)--(7,4)--(7,5)--(6,5)--cycle,black); draw((5,4)--(6,4)); fill((7,5)--(8,5)--(8,6)--(7,6)--cycle,black); draw((7,4)--(8,4)); fill((8,4)--(9,4)--(9,5)--(8,5)--cycle,black); draw((9,4)--(10,4)); label("same",(6.3,2.45),N); label("pattern here",(7.5,1.4),N);[/asy]

1

1988 AJHSME #3

\frac{1}{10}+\frac{2}{20}+\frac{3}{30}=

\text{(A)}\ .1\qquad\text{(B)}\ .123\qquad\text{(C)}\ .2\qquad\text{(D)}\ .3\qquad\text{(E)}\ .6

1

1988 AJHSME #2

8\times .25\times 2\times .125 =

\text{(A)}\ \frac{1}8\qquad\text{(B)}\ \frac{1}4\qquad\text{(C)}\ \frac{1}2\qquad\text{(D)}\ 1\qquad\text{(E)}\ 2

1

1988 AJHSME #1

The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of [asy] draw((-3,0)..(0,3)..(3,0)); draw((-3.5,0)--(-2.5,0)); draw((0,2.5)--(0,3.5)); draw((2.5,0)--(3.5,0)); draw((1.8,1.8)--(2.5,2.5)); draw((-1.8,1.8)--(-2.5,2.5)); draw((0,0)--3*dir(120),EndArrow); label("

10

",(-2.6,0),E); label("

11

",(2.6,0),W);[/asy]

\text{(A)}\ 10.05\qquad\text{(B)}\ 10.15\qquad\text{(C)}\ 10.25\qquad\text{(D)}\ 10.3\qquad\text{(E)}\ 10.6