1988 CMO 第 3 题

Given a finite sequence of real numbers $a_1,a_2,\dots ,a_n$ ( $\ast$ ), we call a segment $a_k,\dots ,a_{k+l-1}$ of the sequence ( $\ast$ ) a “*long*”(Chinese dragon) and $a_k$ “*head*” of the “*long*” if the arithmetic mean of $a_k,\dots ,a_{k+l-1}$ is greater than $1988$ . (especially if a single item $a_m>1988$ , we still regard $a_m$ as a “*long*”). Suppose that there is at least one “*long*” among the sequence ( $\ast$ ), show that the arithmetic mean of all those items of sequence ( $\ast$ ) that could be “*head*” of a certain “*long*” individually is greater than $1988$ .

给定一个有限实数序列 $a_1,a_2,\dots ,a_n$ ( $\ast$ )，如果 $a_k,\dots 的算术平均值，我们将序列 ($\ast$) 的一段$a_k,\dots ,a_{k+l-1}$称为“*long*”(中国龙)，并将“*long*”的$a_k$“*head*” 称为,a_{k+l-1}$ 大于 $1988$ 。 (特别是如果单个项目 $a_m>1988$ ，我们仍然将 $a_m$ 视为“*long*”)。假设序列( $\ast$ )中至少有一个“*long*”，则表明序列( $\ast$ )中所有可能成为某个“*long*”的“*head*”的项的算术平均值大于 $1988$ 。