IMC 2017 · Problem 7

Let $p(x)$ be a nonconstant polynomial with real coefficients. For every positive integer $n$, let

$$q_n(x) = (x + 1)^n p(x) + x^n p(x + 1).$$

Prove that there are only finitely many numbers $n$ such that all roots of $q_n(x)$ are real.