2025 APMO 第 3 题

Let $P(x)$ be a non- constant polynomial with integer coefficients such that $P(0) \neq 0$ . Let $a_{1}, a_{2}, a_{3}, \ldots$ be an infinite sequence of integers such that $P(i - j)$ divides $a_{i} - a_{j}$ for all distinct positive integers $i, j$ . Prove that the sequence $a_{1}, a_{2}, a_{3}, \ldots$ must be constant, that is, $a_{n}$ equals a constant $c$ for all $n$ positive integer.

令 $P(x)$ 为具有整数系数的非常数多项式，使得 $P(0) \neq 0$ 。令 $a_{1}, a_{2}, a_{3}, \ldots$ 为无限整数序列，使得 $P(i - j)$ 能除以 $a_{i} - a_{j}$ 得到所有不同的正整数 $i, j$ 。证明序列 $a_{1}, a_{2}, a_{3}, \ldots$ 必须是常数，即对于所有 $n$ 正整数，$a_{n}$ 等于常数 $c$。