IMC 2011 · Problem 4

Let $f(x)$ be a polynomial with real coefficients of degree $n$. Suppose that $\dfrac{f(k) - f(m)}{k - m}$ is an integer for all integers $0 \le k < m \le n$. Prove that $a - b$ divides $f(a) - f(b)$ for all pairs of distinct integers $a$ and $b$.