2013 IMO Shortlist C5

Let $r$ be a positive integer, and let $a_{0}, a_{1}, \ldots$ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in[m+1, m+r]$ such that $$a_{m}+a_{m+1}+\cdots+a_{m+s}=a_{n}+a_{n+1}+\cdots+a_{n+s}$$ Prove that the sequence is periodic, i. e. there exists some $p \geq 1$ such that $a_{n+p}=a_{n}$ for all $n \geq 0$.

令$r$ 为正整数，并令$a_{0}, a_{1}, \ldots$ 为无限实数序列。假设对于所有非负整数$m$和$s$，存在一个正整数$n \in[m+1, m+r]$使得$$a_{m}+a_{m+1}+\cdots+a_{m+s}=a_{n}+a_{n+1}+\cdots+a_{n+s}$$证明序列是周期性的，i。 e.存在一些$p \geq 1$，使得$a_{n+p}=a_{n}$ 对于所有$n \geq 0$。