APMO 2025 Problem 3

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.