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Difference between revisions of "1997 AIME Problems/Problem 2"

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== Problem ==
 
== Problem ==
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The nine horizontal and nine vertical lines on an <math>8\times8</math> checkerboard form <math>r</math> rectangles, of which <math>s</math> are squares.  The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
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The nine horizontal and nine vertical lines on an <math>8\times8</math> checkerboard form <math>r</math> [[rectangles]], of which <math>s</math> are [[square]]s.  The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
  
 
== Solution ==
 
== Solution ==
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For r, we can choose two out of 9 lines, and 2 out of nine lines again, to get <math>r=(\binom{9}{2})^2=36^2=1296</math>
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To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the chessboard, or <math>{9\choose 2} = 36</math>. Similarily, there are <math>{9\choose 2}</math> ways to pick the vertical sides, giving us <math>r = 1296</math> rectangles.
  
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For s, there are 8^2 unit squares, 7^2 2*2 squares, .... 1^1 8*8 squares. That gives us <math>s=1^2+2^2+\cdots+8^2=\dfrac{8*9*17}{6}=12*17=204</math>
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For <math>s</math>, there are <math>8^2</math> [[unit square]]s, <math>7^2</math> of the <math>2\times2</math> squares, and so on until <math>1^2</math> of the <math>8\times 8</math> squares. Using the sum of squares formula, that gives us <math>s=1^2+2^2+\cdots+8^2=\dfrac{(8)(8+1)(2\cdot8+1}{6}=12*17=204</math>.
  
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<math>\dfrac{204}{1296}=\dfrac{17}{108}</math>
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Thus <math>\frac rs = \dfrac{204}{1296}=\dfrac{17}{108}</math>, and <math>m+n=\boxed{125}</math>.
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<math>m+n=125</math>
 
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1997|num-b=1|num-a=3}}
 
{{AIME box|year=1997|num-b=1|num-a=3}}
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[[Category:Intermediate Combinatorics Problems]]

Revision as of 17:33, 21 November 2007

Problem

The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the chessboard, or ${9\choose 2} = 36$. Similarily, there are ${9\choose 2}$ ways to pick the vertical sides, giving us $r = 1296$ rectangles.

For $s$, there are $8^2$ unit squares, $7^2$ of the $2\times2$ squares, and so on until $1^2$ of the $8\times 8$ squares. Using the sum of squares formula, that gives us $s=1^2+2^2+\cdots+8^2=\dfrac{(8)(8+1)(2\cdot8+1}{6}=12*17=204$.

Thus $\frac rs = \dfrac{204}{1296}=\dfrac{17}{108}$, and $m+n=\boxed{125}$.

See also

1997 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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