Question 4
Let $\mathbb Z$ be the integers. Find all polynomials $P(x)$ with integer coefficients such that, for every infinite sequence $a_1,a_2,\ldots$ containing each integer exactly once, there are indices $i\lt j$ and an integer $k$ satisfying
$$a_i+a_{i+1}+\cdots+a_j=P(k).$$
$$a_i+a_{i+1}+\cdots+a_j=P(k).$$