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gcd(f(m) + n, f(n) + m) bounded for m != n

Source: IMO 2015 Shortlist, N7

July 7, 2016
functionnumber theorygreatest common divisorIMO Shortlist

Problem Statement

Let Z>0\mathbb{Z}_{>0} denote the set of positive integers. For any positive integer kk, a function f:Z>0→Z>0f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0} is called kk-good if gcd⁡(f(m)+n,f(n)+m)≤k\gcd(f(m) + n, f(n) + m) \le k for all m≠nm \neq n. Find all kk such that there exists a kk-good function.
Proposed by James Rickards, Canada
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