2000 AIME I Problems/Problem 4
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.
Call the squares' side lengths from smallest to largest , and let represent the dimensions of the rectangle.
The picture shows that
With a bit of trial and error and some arithmetic, we can use these equations to find that ; we can guess that . Then solving gives , , , which gives us . These numbers are relatively prime, as desired. (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.) The perimeter is .
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