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very nice combinatorics with NT

Source: III Caucasus Mathematical Olympiad

March 17, 2018
combinatoricsnumber theory

Problem Statement

For 2n2n positive integers a matching (i.e. dividing them into nn pairs) is called {\it non-square} if the product of two numbers in each pair is not a perfect square. Prove that if there is a non-square matching, then there are at least n!n! non-square matchings. (By n!n! denote the product 1⋅2⋅3⋅…⋅n1\cdot 2\cdot 3\cdot \ldots \cdot n.)
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