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IMC 2011 Day 2, Problem 4

Source:

August 1, 2011
algebrapolynomialinductionIMCcollege contests

Problem Statement

Let ff be a polynomial with real coefficients of degree nn. Suppose that f(x)−f(y)x−y\displaystyle \frac{f(x)-f(y)}{x-y} is an integer for all 0≤x<y≤n0 \leq x<y \leq n. Prove that a−b∣f(a)−f(b)a-b | f(a)-f(b) for all distinct integers a,ba,b.
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