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> checkerboard form <math>r</math> rectangles, of which <math>s</math> are | + | 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 == | ||
− | + | 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. | |
− | For s, there are 8^2 unit | + | 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>. |
− | <math>\dfrac{204}{1296}=\dfrac{17}{108}</math> | + | 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 18:33, 21 November 2007
Problem
The nine horizontal and nine vertical lines on an checkerboard form
rectangles, of which
are squares. The number
can be written in the form
where
and
are relatively prime positive integers. Find
Solution
To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the chessboard, or . Similarily, there are
ways to pick the vertical sides, giving us
rectangles.
For , there are
unit squares,
of the
squares, and so on until
of the
squares. Using the sum of squares formula, that gives us
.
Thus , and
.
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 |