1983 AIME Problems/Problem 12
Problem
The length of diameter is a two digit integer. Reversing the digits gives the length of a perpendicular chord . The distance from their intersection point to the center is a positive rational number. Determine the length of .
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Solution
Let and . It follows that and . Applying the Pythagorean Theorem on and , .
Because is a positive rational number, the quantity cannot contain any square roots. Therefore, must equal eleven and must be a perfect square (since ). The only pair that satisfies this condition is , so our answer is .