2023 INMO Problems/Problem 1

PROBLEM

Let S be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs (x, y) in S × S so that the product xy is a perfect square. Prove that one can find at least four distinct elements in S so that none of their pairwise products is a perfect square. Note: As an example, if S = {1, 2, 4}, there are exactly five such ordered pairs: (1, 1), (1, 4), (2, 2), (4, 1), and (4, 4).

==SOLUTION==hi