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FInd all f such that f(n!)=f(n)! and m-n divides f(m)-f(n)

Source: Balkan MO 2012 - Problem 4

April 28, 2012
functionAMCUSA(J)MOUSAMOinequalitiesfunctional equationBalkan

Problem Statement

Let Z+\mathbb{Z}^+ be the set of positive integers. Find all functions f:Z+→Z+f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+ such that the following conditions both hold: (i) f(n!)=f(n)!f(n!)=f(n)! for every positive integer nn, (ii) m−nm-n divides f(m)−f(n)f(m)-f(n) whenever mm and nn are different positive integers.
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