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Integer square root given fraction identity

Source: Polish Junior Olympiad 2021 Final Round Problem 1, https://omj.edu.pl

March 21, 2021
positive integerssquare rootsalgebra

Problem Statement

Positive integers aa, bb an nn satisfy ab=a2+n2b2+n2. \frac{a}{b}=\frac{a^2+n^2}{b^2+n^2}. Prove that ab\sqrt{ab} is an integer.
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