2010 IMO Shortlist A6

Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n))=f(n)+1$ and $g(f(n))=$ $g(n)+1$ hold for all positive integers. Prove that $f(n)=g(n)$ for all positive integer $n$. (Germany)

假设 $f$ 和 $g$ 是在正整数集合上定义并取正整数值的两个函数。还假设方程 $f(g(n))=f(n)+1$ 和 $g(f(n))=$ $g(n)+1$ 对于所有正整数都成立。证明对于所有正整数 $n$，$f(n)=g(n)$。 (德国)