2012 IMO Shortlist A6

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^{m}$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2 k}(n)=n+k$, and let $k_{n}$ be the smallest such $k$. Prove that the sequence $k_{1}, k_{2}, \ldots$ is unbounded.

令$f: \mathbb{N} \rightarrow \mathbb{N}$ 为函数，并令$f^{m}$ 为$f$ 应用$m$ 次。假设对于每个 $n \in \mathbb{N}$ 都存在一个 $k \in \mathbb{N}$ 使得 $f^{2 k}(n)=n+k$，并令 $k_{n}$ 为最小的 $k$。证明序列 $k_{1}, k_{2}, \ldots$ 是无界的。