2000 AIME I Problems/Problem 4
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[hide]Problem
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
Solution 1
Call the squares' side lengths from smallest to largest , and let represent the dimensions of the rectangle.
The picture shows that
Expressing all terms 3 to 9 in terms of and and substituting their expanded forms into the previous equation will give the expression .
We can guess that . (If we started with odd, the resulting sides would not be integers and we would need to scale up by a factor of to make them integers; if we started with even, the resulting dimensions would not be relatively prime and we would need to scale down.) Then solving gives , , , which gives us . These numbers are relatively prime, as desired. The perimeter is .
Solution 2 Length-chasing (Angle-chasing but for side lengths)
See also
2000 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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