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Functional Equation

Source: SAMO 2011 Senior Round 3 Problem 5

September 7, 2011
functioninequalitiesnumber theoryrelatively primenumber theory unsolved

Problem Statement

Let N0\mathbb{N}_0 denote the set of all nonnegative integers. Determine all functions f:N0→N0f:\mathbb{N}_0\to\mathbb{N}_0 with the following two properties:
[*] 0≤f(x)≤x20\le f(x)\le x^2 for all x∈N0x\in\mathbb{N}_0
[*] x−yx-y divides f(x)−f(y)f(x)-f(y) for all x,y∈N0x,y\in\mathbb{N}_0 with x>yx>y
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