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Difference between revisions of "1994 AJHSME Problems/Problem 12"
 
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==Problem==
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== Problem 12 ==
  
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A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: ''The product of the month times the day equals the last two digits of the year''.  For example, 1956 is a lucky year because it has the date 7/8/56 and <math>7\times 8 = 56</math>.  Which of the following is NOT a lucky year?
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Each of the three large squares shown below is the same size.  Segments that intersect the sides of the squares intersect at the midpoints of the sides.  How do the shaded areas of these squares compare?
  
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<math>\text{(A)}\ 1990 \qquad \text{(B)}\ 1991 \qquad \text{(C)}\ 1992 \qquad \text{(D)}\ 1993 \qquad \text{(E)}\ 1994</math>
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<center>
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<asy>
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unitsize(36);
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fill((0,0)--(1,0)--(1,1)--cycle,gray);
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fill((1,1)--(1,2)--(2,2)--cycle,gray);
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draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
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draw((1,0)--(1,2));
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draw((0,0)--(2,2));
  
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==Solution==
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fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,gray);
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<math>90=(3)(30)</math> which corresponds to March 30.
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draw((3,0)--(5,0)--(5,2)--(3,2)--cycle);
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draw((4,0)--(4,2));
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draw((3,1)--(5,1));
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fill((6,1)--(6.5,0.5)--(7,1)--(7.5,0.5)--(8,1)--(7.5,1.5)--(7,1)--(6.5,1.5)--cycle,gray);
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draw((6,0)--(8,0)--(8,2)--(6,2)--cycle);
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draw((6,0)--(8,2));
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draw((6,2)--(8,0));
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draw((7,0)--(6,1)--(7,2)--(8,1)--cycle);
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label("$I$",(1,2),N);
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label("$II$",(4,2),N);
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label("$III$",(7,2),N);
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</asy>
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</center>
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<math>\text{(A)}\ \text{The shaded areas in all three are equal.}</math>
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<math>\text{(B)}\ \text{Only the shaded areas of }I\text{ and }II\text{ are equal.}</math>
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<math>\text{(C)}\ \text{Only the shaded areas of }I\text{ and }III\text{ are equal.}</math>
  
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<math>91=(7)(13)</math> which corresponds to July 13.
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<math>\text{(D)}\ \text{Only the shaded areas of }II\text{ and }III\text{ are equal.}</math>
  
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<math>92=(4)(23)</math> which corresponds to April 23.
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<math>\text{(E)}\ \text{The shaded areas of }I, II\text{ and }III\text{ are all different.}</math>
  
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<math>93=(3)(31)</math> which corresponds to March 31.
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==Solution==
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Square II clearly has <math>1/4</math> shaded. Partitioning square I into eight right triangles also shows <math>1/4</math> of it is shaded. Lastly, square III can be partitioned into sixteen triangles, and because four are shaded, <math>1/4</math> of the total square is shaded. <math>\boxed{\text{(A)}}</math>.
  
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<math>94</math> can only be factored into <math>(2)(47)</math>, which does not correspond to any date. The only year that is not lucky is <math>\boxed{\text{(E)}\ 1994}</math>.
 
 
==See Also==
 
==See Also==
 
{{AJHSME box|year=1994|num-b=11|num-a=13}}
 
{{AJHSME box|year=1994|num-b=11|num-a=13}}
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{{MAA Notice}}

Latest revision as of 00:13, 5 July 2013

Problem 12

Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare?

[asy] unitsize(36); fill((0,0)--(1,0)--(1,1)--cycle,gray); fill((1,1)--(1,2)--(2,2)--cycle,gray); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,0)--(2,2)); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,gray); draw((3,0)--(5,0)--(5,2)--(3,2)--cycle); draw((4,0)--(4,2)); draw((3,1)--(5,1)); fill((6,1)--(6.5,0.5)--(7,1)--(7.5,0.5)--(8,1)--(7.5,1.5)--(7,1)--(6.5,1.5)--cycle,gray); draw((6,0)--(8,0)--(8,2)--(6,2)--cycle); draw((6,0)--(8,2)); draw((6,2)--(8,0)); draw((7,0)--(6,1)--(7,2)--(8,1)--cycle); label("$I$",(1,2),N); label("$II$",(4,2),N); label("$III$",(7,2),N); [/asy]

$\text{(A)}\ \text{The shaded areas in all three are equal.}$

$\text{(B)}\ \text{Only the shaded areas of }I\text{ and }II\text{ are equal.}$

$\text{(C)}\ \text{Only the shaded areas of }I\text{ and }III\text{ are equal.}$

$\text{(D)}\ \text{Only the shaded areas of }II\text{ and }III\text{ are equal.}$

$\text{(E)}\ \text{The shaded areas of }I, II\text{ and }III\text{ are all different.}$

Solution

Square II clearly has $1/4$ shaded. Partitioning square I into eight right triangles also shows $1/4$ of it is shaded. Lastly, square III can be partitioned into sixteen triangles, and because four are shaded, $1/4$ of the total square is shaded. $\boxed{\text{(A)}}$.

See Also

1994 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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. AMC logo.png

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