MathDB

1991 AMC 8

Part of AMC 8

Subcontests

(25)
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AJHSME 1991 Problem 25

An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black?
[asy] unitsize(36); fill((0,0)--(2,0)--(1,sqrt(3))--cycle,gray); draw((0,0)--(2,0)--(1,sqrt(3))--cycle,linewidth(1)); fill((4,0)--(6,0)--(5,sqrt(3))--cycle,gray); fill((5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--cycle,white); draw((5,sqrt(3))--(4,0)--(5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--(5,0)--(6,0)--cycle,linewidth(1)); fill((8,0)--(10,0)--(9,sqrt(3))--cycle,gray); fill((9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--cycle,white); fill((17/2,0)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--cycle,white); fill((9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--cycle,white); fill((19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--cycle,white); draw((9,sqrt(3))--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--(9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--(17/2,0)--(33/4,sqrt(3)/4)--(8,0)--(9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--(9,0)--(19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--(19/2,0)--(10,0)--cycle,linewidth(1)); label("Change 1",(3,3*sqrt(3)/4),N); label("⟹\Longrightarrow ",(3,5*sqrt(3)/8),S); label("Change 2",(7,3*sqrt(3)/4),N); label("⟹\Longrightarrow ",(7,5*sqrt(3)/8),S); [/asy]
(A) 11024(B) 1564(C) 2431024(D) 14(E) 81256\text{(A)}\ \frac{1}{1024} \qquad \text{(B)}\ \frac{15}{64} \qquad \text{(C)}\ \frac{243}{1024} \qquad \text{(D)}\ \frac{1}{4} \qquad \text{(E)}\ \frac{81}{256}
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AJHSME 1991 Problem 22

Each spinner is divided into 33 equal parts. The results obtained from spinning the two spinners are multiplied. What is the probability that this product is an even number?
[asy] draw(circle((0,0),2)); draw(circle((5,0),2)); draw((0,0)--(sqrt(3),1)); draw((0,0)--(-sqrt(3),1)); draw((0,0)--(0,-2)); draw((5,0)--(5+sqrt(3),1)); draw((5,0)--(5-sqrt(3),1)); draw((5,0)--(5,-2)); fill((0,5/3)--(2/3,7/3)--(1/3,7/3)--(1/3,3)--(-1/3,3)--(-1/3,7/3)--(-2/3,7/3)--cycle,black); fill((5,5/3)--(17/3,7/3)--(16/3,7/3)--(16/3,3)--(14/3,3)--(14/3,7/3)--(13/3,7/3)--cycle,black); label("11",(0,1/2),N); label("22",(sqrt(3)/4,-1/4),ESE); label("33",(-sqrt(3)/4,-1/4),WSW); label("44",(5,1/2),N); label("55",(5+sqrt(3)/4,-1/4),ESE); label("66",(5-sqrt(3)/4,-1/4),WSW); [/asy]
(A) 13(B) 12(C) 23(D) 79(E) 1\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ \frac{7}{9} \qquad \text{(E)}\ 1
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AJHSME 1991 Problem 18

The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for 55 years or more?
[asy] for(int a=1; a<11; ++a) { draw((a,0)--(a,-.5)); } draw((0,10.5)--(0,0)--(10.5,0)); label("11",(1,-.5),S); label("22",(2,-.5),S); label("33",(3,-.5),S); label("44",(4,-.5),S); label("55",(5,-.5),S); label("66",(6,-.5),S); label("77",(7,-.5),S); label("88",(8,-.5),S); label("99",(9,-.5),S); label("1010",(10,-.5),S); label("Number of years with company",(5.5,-2),S); label("X",(1,0),N); label("X",(1,1),N); label("X",(1,2),N); label("X",(1,3),N); label("X",(1,4),N); label("X",(2,0),N); label("X",(2,1),N); label("X",(2,2),N); label("X",(2,3),N); label("X",(2,4),N); label("X",(3,0),N); label("X",(3,1),N); label("X",(3,2),N); label("X",(3,3),N); label("X",(3,4),N); label("X",(3,5),N); label("X",(3,6),N); label("X",(3,7),N); label("X",(4,0),N); label("X",(4,1),N); label("X",(4,2),N); label("X",(5,0),N); label("X",(5,1),N); label("X",(6,0),N); label("X",(6,1),N); label("X",(7,0),N); label("X",(7,1),N); label("X",(8,0),N); label("X",(9,0),N); label("X",(10,0),N); label("Gauss Company",(5.5,10),N); [/asy]
(A) 9%(B) 2313%(C) 30%(D) 4267%(E) 50%\text{(A)}\ 9\% \qquad \text{(B)}\ 23\frac{1}{3}\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 42\frac{6}{7}\% \qquad \text{(E)}\ 50\%
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AJHSME 1991 Problem 16

The 1616 squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
[*]fold the top half over the bottom half [*]fold the bottom half over the top half [*]fold the right half over the left half [*]fold the left half over the right half. Which numbered square is on top after step 44?
[asy] unitsize(18); for(int a=0; a<5; ++a) { draw((a,0)--(a,4)); } for(int b=0; b<5; ++b) { draw((0,b)--(4,b)); } label("11",(0.5,3.1),N); label("22",(1.5,3.1),N); label("33",(2.5,3.1),N); label("44",(3.5,3.1),N); label("55",(0.5,2.1),N); label("66",(1.5,2.1),N); label("77",(2.5,2.1),N); label("88",(3.5,2.1),N); label("99",(0.5,1.1),N); label("1010",(1.5,1.1),N); label("1111",(2.5,1.1),N); label("1212",(3.5,1.1),N); label("1313",(0.5,0.1),N); label("1414",(1.5,0.1),N); label("1515",(2.5,0.1),N); label("1616",(3.5,0.1),N); [/asy]
(A) 1(B) 9(C) 10(D) 14(E) 16\text{(A)}\ 1 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16
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AJHSME 1991 Problem 15

All six sides of a rectangular solid were rectangles. A one-foot cube was cut out of the rectangular solid as shown. The total number of square feet in the surface of the new solid is how many more or less than that of the original solid?
[asy] unitsize(20); draw((0,0)--(1,0)--(1,3)--(0,3)--cycle); draw((1,0)--(1+9*sqrt(3)/2,9/2)--(1+9*sqrt(3)/2,15/2)--(1+5*sqrt(3)/2,11/2)--(1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),4)--(1+2*sqrt(3),5)--(1,3)); draw((0,3)--(2*sqrt(3),5)--(1+2*sqrt(3),5)); draw((1+9*sqrt(3)/2,15/2)--(9*sqrt(3)/2,15/2)--(5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,5)); draw((1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),9/2)); draw((1+5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,11/2)); label("1′1'",(.5,0),S); label("3′3'",(1,1.5),E); label("9′9'",(1+9*sqrt(3)/4,9/4),S); label("1′1'",(1+9*sqrt(3)/4,17/4),S); label("1′1'",(1+5*sqrt(3)/2,5),E);label("1′1'",(1/2+5*sqrt(3)/2,11/2),S); [/asy]
(A) 2 less(B) 1 less(C) the same(D) 1 more(E) 2 more\text{(A)}\ 2\text{ less} \qquad \text{(B)}\ 1\text{ less} \qquad \text{(C)}\ \text{the same} \qquad \text{(D)}\ 1\text{ more} \qquad \text{(E)}\ 2\text{ more}
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AJHSME 1991 Problem 5

A "domino" is made up of two small squares: [asy] unitsize(12); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((1,1)--(2,1)--(2,0)--(1,0)); [/asy] Which of the "checkerboards" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes?
[asy] unitsize(12); fill((0,0)--(1,0)--(1,1)--(0,1)--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((3,1)--(4,1)--(4,2)--(3,2)--cycle,black); fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,black); fill((2,2)--(2,3)--(3,3)--(3,2)--cycle,black); draw((0,0)--(0,3)--(4,3)--(4,0)--cycle); draw((6,0)--(11,0)--(11,3)--(6,3)--cycle); fill((6,0)--(7,0)--(7,1)--(6,1)--cycle,black); fill((8,0)--(9,0)--(9,1)--(8,1)--cycle,black); fill((10,0)--(11,0)--(11,1)--(10,1)--cycle,black); fill((7,1)--(7,2)--(8,2)--(8,1)--cycle,black); fill((9,1)--(9,2)--(10,2)--(10,1)--cycle,black); fill((6,2)--(6,3)--(7,3)--(7,2)--cycle,black); fill((8,2)--(8,3)--(9,3)--(9,2)--cycle,black); fill((10,2)--(10,3)--(11,3)--(11,2)--cycle,black); draw((13,-1)--(13,3)--(17,3)--(17,-1)--cycle); fill((13,3)--(14,3)--(14,2)--(13,2)--cycle,black); fill((15,3)--(16,3)--(16,2)--(15,2)--cycle,black); fill((14,2)--(15,2)--(15,1)--(14,1)--cycle,black); fill((16,2)--(17,2)--(17,1)--(16,1)--cycle,black); fill((13,1)--(14,1)--(14,0)--(13,0)--cycle,black); fill((15,1)--(16,1)--(16,0)--(15,0)--cycle,black); fill((14,0)--(15,0)--(15,-1)--(14,-1)--cycle,black); fill((16,0)--(17,0)--(17,-1)--(16,-1)--cycle,black); draw((19,3)--(24,3)--(24,-1)--(19,-1)--cycle,black); fill((19,3)--(20,3)--(20,2)--(19,2)--cycle,black); fill((21,3)--(22,3)--(22,2)--(21,2)--cycle,black); fill((23,3)--(24,3)--(24,2)--(23,2)--cycle,black); fill((20,2)--(21,2)--(21,1)--(20,1)--cycle,black); fill((22,2)--(23,2)--(23,1)--(22,1)--cycle,black); fill((19,1)--(20,1)--(20,0)--(19,0)--cycle,black); fill((21,1)--(22,1)--(22,0)--(21,0)--cycle,black); fill((23,1)--(24,1)--(24,0)--(23,0)--cycle,black); fill((20,0)--(21,0)--(21,-1)--(20,-1)--cycle,black); fill((22,0)--(23,0)--(23,-1)--(22,-1)--cycle,black); draw((26,3)--(29,3)--(29,-3)--(26,-3)--cycle); fill((26,3)--(27,3)--(27,2)--(26,2)--cycle,black); fill((28,3)--(29,3)--(29,2)--(28,2)--cycle,black); fill((27,2)--(28,2)--(28,1)--(27,1)--cycle,black); fill((26,1)--(27,1)--(27,0)--(26,0)--cycle,black); fill((28,1)--(29,1)--(29,0)--(28,0)--cycle,black); fill((27,0)--(28,0)--(28,-1)--(27,-1)--cycle,black); fill((26,-1)--(27,-1)--(27,-2)--(26,-2)--cycle,black); fill((28,-1)--(29,-1)--(29,-2)--(28,-2)--cycle,black); fill((27,-2)--(28,-2)--(28,-3)--(27,-3)--cycle,black); [/asy]
(A) 3×4(B) 3×5(C) 4×4(D) 4×5(E) 6×3\text{(A)}\ 3\times 4 \qquad \text{(B)}\ 3\times 5 \qquad \text{(C)}\ 4\times 4 \qquad \text{(D)}\ 4\times 5 \qquad \text{(E)}\ 6\times 3
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