1993 IMO 第 6 题

(GER 1) ${ }^{\mathrm{IMO} 5}$ Let $\mathbb{N}=\{1,2,3, \ldots\}$. Determine whether there exists a strictly increasing function $f: \mathbb{N} \rightarrow \mathbb{N}$ with the following properties: $$\begin{aligned} f(1) & =2 \\ f(f(n)) & =f(n)+n \quad(n \in \mathbb{N}) \end{aligned}$$

(GER 1) ${ }^{\mathrm{IMO} 5}$ 令 $\mathbb{N}=\{1,2,3, \ldots\}$。判断是否存在严格递增函数$f: \mathbb{N} \rightarrow \mathbb{N}$，具有以下性质： $$\begin{aligned} f(1) & =2 \\ f(f(n)) & =f(n)+n \quad(n \in \mathbb{N}) \end{aligned}$$