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

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== Problem ==
 
== Problem ==
The nine horizontal and nine vertical lines on an <math>8\times8</math> checkeboard 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 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>
  
 
== Solution ==
 
== Solution ==

Revision as of 13:12, 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

For r, we can choose two out of 9 lines, and 2 out of nine lines again, to get $r=(\binom{9}{2})^2=36^2=1296$

For s, there are 8^2 unit squares, 7^2 2*2 squares, .... 1^1 8*8 squares. That gives us $s=1^2+2^2+\cdots+8^2=\dfrac{8*9*17}{6}=12*17=204$

$\dfrac{204}{1296}=\dfrac{17}{108}$

$m+n=125$

See also

1997 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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All AIME Problems and Solutions
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