2011 IMO Shortlist A4

Determine all pairs $(f, g)$ of functions from the set of positive integers to itself that satisfy $$f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1$$ for every positive integer $n$. Here, $f^{k}(n)$ means $\underbrace{f(f(\ldots f}_{k}(n) \ldots))$.

对于每个正整数 $n$，确定从正整数集合到其本身满足 $$f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1$$ 的所有函数对 $(f, g)$。这里，$f^{k}(n)$ 表示$\underbrace{f(f(\ldots f}_{k}(n) \ldots))$。