2010 IMO Shortlist A7

Let $a_{1}, \ldots, a_{r}$ be positive real numbers. For $n>r$, we inductively define $$a_{n}=\max _{1 \leq k \leq n-1}\left(a_{k}+a_{n-k}\right)$$ Prove that there exist positive integers $\ell \leq r$ and $N$ such that $a_{n}=a_{n-\ell}+a_{\ell}$ for all $n \geq N$. (Iran)

令 $a_{1}, \ldots, a_{r}$ 为正实数。对于$n>r$，我们归纳定义$$a_{n}=\max _{1 \leq k \leq n-1}\left(a_{k}+a_{n-k}\right)$$证明存在正整数$\ell \leq r$和$N$，使得对于所有$n \geq N$，$a_{n}=a_{n-\ell}+a_{\ell}$。 (伊朗)