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IMO Shortlist 2009 - Problem N3

Source:

July 5, 2010
functionalgebramodular arithmeticnumber theoryDivisibilityIMO Shortlist

Problem Statement

Let ff be a non-constant function from the set of positive integers into the set of positive integer, such that a−ba-b divides f(a)−f(b)f(a)-f(b) for all distinct positive integers aa, bb. Prove that there exist infinitely many primes pp such that pp divides f(c)f(c) for some positive integer cc.
Proposed by Juhan Aru, Estonia
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