Difference between revisions of "1997 AIME Problems/Problem 2"
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== Solution == | == Solution == | ||
+ | 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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+ | 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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+ | <math>\dfrac{204}{1296}=\dfrac{17}{108}</math> | ||
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+ | <math>m+n=125</math> | ||
== See also == | == See also == | ||
* [[1997 AIME Problems]] | * [[1997 AIME Problems]] |
Revision as of 11:21, 11 October 2007
Problem
The nine horizontal and nine vertical lines on an checkeboard form
rectangles, of which
are squares. The number
can be written in the form
where
and
are relatively prime positive integers. Find
Solution
For r, we can choose two out of 9 lines, and 2 out of nine lines again, to get
For s, there are 8^2 unit squares, 7^2 2*2 squares, .... 1^1 8*8 squares. That gives us