Problem

Source: 2019 Ukraine TST 2.2

Tags: algebra, polynomial, divisible



Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property: for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer. Prove that this polynomial is divisible by $x$.