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Divisibility holds for all naturals

Source: 2018 Balkan MO Shortlist N5

May 22, 2019
number theoryBalkan MO Shortlist

Problem Statement

Let x,yx,y be positive integers. If for each positive integer nn we have that (ny)2+1∣xφ(n)−1.(ny)^2+1\mid x^{\varphi(n)}-1. Prove that x=1x=1.
(Silouanos Brazitikos, Greece)
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