2003 AIME II Problems/Problem 10
Two positive integers differ by . The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?
Call the two integers and , so we have . Square both sides to get . Thus, must be a square, so we have , and . The sum of these two factors is , so they must both be even. To maximize , we want to maximixe , so we let it equal and the other factor , but solving gives , which is already a perfect square, so we have to keep going. In order to keep both factors even, we let the larger one equal and the other , which gives . This checks, so the solution is .
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