APMO 2020 P4
Posted: Thu Dec 03, 2020 9:23 pm
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 $a_1$, $a_2$, $\cdots$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\cdots +a_j = P(k)$.
For any infinite sequence $a_1$, $a_2$, $\cdots$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\cdots +a_j = P(k)$.