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Product of pairs is a perfect square

Source: 2024 Azerbaijan BMO TST

July 18, 2024
combinatoricsAZE BMO TSTTST

Problem Statement

Let nn be a positive integer. Using the integers from 11 to 4n4n inclusive, pairs are to be formed such that the product of the numbers in each pair is a perfect square. Each number can be part of at most one pair, and the two numbers in each pair must be different. Determine, for each nn, the maximum number of pairs that can be formed.