2015 CMO 第 6 题

Let $a_1,a_2,...$ be a sequence of non-negative integers such that for any $m,n$ $$\sum_{i=1}^{2m} a_{in} \leq m.$$ Show that there exist $k,d$ such that $$\sum_{i=1}^{2k} a_{id} = k-2014.$$

令 $a_1,a_2,...$ 为非负整数序列，使得对于任何 $m,n$ $$\sum_{i=1}^{2m} a_{in} \leq m.$$ 表明存在 $k,d$ 使得 $$\sum_{i=1}^{2k} a_{id} = k-2014.$$