MathDB

5

Part of 2011 South africa National Olympiad

Problems(1)

Functional Equation

Source: SAMO 2011 Senior Round 3 Problem 5

9/7/2011
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
functioninequalitiesnumber theoryrelatively primenumber theory unsolved