2005 Indonesia MO Problems/Problem 3
Let and be positive integers such that is an integer.
(a) Prove that is rational.
(b) Prove that is a positive integer.
Let , where all variables are integers. Rearranging the expression results in .
Squaring both sides results in , and rearranging terms results in .
Squaring both sides results in . Solving for results in . Since the right side is rational, the left side must be rational. Therefore, is rational, and since is a positive integer, must be a positive integer.
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