2017 IMO Shortlist A4

A sequence of real numbers $a_{1}, a_{2}, \ldots$ satisfies the relation $$a_{n}=-\max _{i+j=n}\left(a_{i}+a_{j}\right) \quad \text { for all } n>2017$$ Prove that this sequence is bounded, i.e., there is a constant $M$ such that $\left|a_{n}\right| \leq M$ for all positive integers $n$.

实数序列 $a_{1}, a_{2}, \ldots$ 满足关系 $$a_{n}=-\max _{i+j=n}\left(a_{i}+a_{j}\right) \quad \text { for all } n>2017$$ 证明该序列是有界的，即存在一个常数 $M$ 使得$\left|a_{n}\right| \leq M$ 表示所有正整数 $n$。