2010 IMO 第 6 题

Let $a_{1},a_{2},a_{3},\ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that

$$a_{n} = \max \{a_{k} + a_{n - k}\mid 1\leq k\leq n - 1\} \mathrm{~for~all~}n > s.$$

Prove there exist positive integers $\ell \leq s$ and $N$ , such that

$$a_{n} = a_{\ell} + a_{n - \ell}\mathrm{~for~all~}n\geq N.$$

令 $a_{1},a_{2},a_{3},\ldots$ 为正实数序列，$s$ 为正整数，这样

$$a_{n} = \max \{a_{k} + a_{n - k}\mid 1\leq k\leq n - 1\} \mathrm{~for~all~}n > s.$$

证明存在正整数 $\ell \leq s$ 和 $N$ ，使得

$$a_{n} = a_{\ell} + a_{n - \ell}\mathrm{~for~all~}n\geq N.$$