1994 USAMO 第 1 题

Let $\, k_1 < k_2 < k_3 <\cdots\,$, be positive integers, no two consecutive, and let $\, s_m = k_1+k_2+\cdots+k_m\,$, for $\, m = 1,2,3,\ldots\;\;$. Prove that, for each positive integer $n$, the interval $\, [s_n, s_{n+1})\,$, contains at least one perfect square.

令 $\, k_1 < k_2 < k_3 <\cdots\,$ 为正整数，不连续两个，并令 $\, s_m = k_1+k_2+\cdots+k_m\,$，对于 $\, m = 1,2,3,\ldots\;\;$。证明，对于每个正整数 $n$，区间 $\, [s_n, s_{n+1})\,$ 至少包含一个完全平方数。