1995 USAMO 第 4 题

Suppose $q_1,q_2,...$ is an infinite sequence of integers satisfying the following two conditions:

(a) $m - n$ divides $q_m - q_n$ for $m>n \geq 0$

(b) There is a polynomial $P$ such that $|q_n|<P(n)$ for all $n$.

Prove that there is a polynomial $Q$ such that $q_n = Q(n)$ for each $n$.

假设 $q_1,q_2,...$ 是满足以下两个条件的无限整数序列：

(a) $m - n$ 除 $q_m - q_n$ 得 $m>n \geq 0$

(b) 存在一个多项式$P$，对于所有$n$，$|q_n|<P(n)$。

证明对于每个 $n$ 存在一个多项式 $Q$，使得 $q_n = Q(n)$。