2020 APMO 第 4 题

Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:

For any infinite sequence $a_{1}, a_{2}, \ldots$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i<j$ and an integer $k$ such that $a_{i}+a_{i+1}+\cdots+a_{j}=P(k)$.

Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:

对于任何整数的无限序列 $a_{1}, a_{2}, \ldots$，其中 $\mathbb{Z}$ 中的每个整数只出现一次，存在索引 $i<j$ 和整数 $k$，使得 $a_{i}+a_{i+1}+\cdots+a_{j}=P(k)$。