2011 IMO Shortlist N5

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that for any two integers $m$ and $n$, the difference $f(m)-f(n)$ is divisible by $f(m-n)$. Prove that for all integers $m, n$ with $f(m) \leq f(n)$ the number $f(n)$ is divisible by $f(m)$.

令 $f$ 为从整数集到正整数集的函数。假设对于任意两个整数 $m$ 和 $n$，差值 $f(m)-f(n)$ 可被 $f(m-n)$ 整除。证明对于所有整数 $m, n$ 和 $f(m) \leq f(n)$，数字 $f(n)$ 可以被 $f(m)$ 整除。