2021 APMO 第 2 题

For a polynomial $P$ and a positive integer $n$, define $P_{n}$ as the number of positive integer pairs $(a, b)$ such that $a<b \leq n$ and $|P(a)|-|P(b)|$ is divisible by $n$.

Determine all polynomial $P$ with integer coefficients such that for all positive integers $n, P_{n} \leq 2021$.

对于多项式 $P$ 和正整数 $n$，将 $P_{n}$ 定义为正整数对 $(a, b)$ 的数量，使得 $a<b \leq n$ 且 $|P(a)|-|P(b)|$ 可被 $n$ 整除。

Determine all polynomial $P$ with integer coefficients such that for all positive integers $n, P_{n} \leq 2021$.