Problem:
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
Answer Choices:
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Solution:
Let be the length of the short side of the rectangle, and let be the length of the long side. Then the length of the diagonal is , and
Let be the requested squared ratio. Then , so . By the quadratic formula, the positive solution is .
The problems on this page are the property of the MAA's American Mathematics Competitions