Problem

Source: INAMO 2015 Shortlist A7

Tags: integer polynomials, algebra, polynomial



Suppose $P(n) $ is a nonconstant polynomial where all of its coefficients are nonnegative integers such that \[ \sum_{i=1}^n P(i) | nP(n+1) \]for every $n \in \mathbb{N}$. Prove that there exists an integer $k \ge 0$ such that \[ P(n) = \binom{n+k}{n-1} P(1) \]for every $n \in \mathbb{N}$.