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Old ISL resurrected

Source: STEMS Mathematics 2023 Category B P4

January 8, 2023
combinatorics

Problem Statement

Define a positive integer nn to be a fake square if either n=1n = 1 or nn can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer k⩾2k \geqslant 2, there exist distinct positive integers a1a_1, a2,⋯ ,aka_2, \cdots, a_k such that the polynomial (x+a1)(x+a2)⋯(x+ak)(x+a_1)(x+a_2) \cdots (x+a_k) takes ‘fake square’ values for all x=1,2,⋯ ,2023x = 1,2,\cdots,2023.
Proposed by Prof. Aditya Karnataki
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