International Mathematics Competition for University Students
July 31 – August 6 2017, Blagoevgrad, Bulgaria
7. Let $p(x)$ be a nonconstant polynomial with real coefficients. For every positive integer~$n$, let $$q_n(x) = (x+1)^np(x)+x^n p(x+1) .$$
Prove that there are only finitely many numbers $n$ such that all roots of $q_n(x)$ are real.
Proposed by: Alexandr Bolbot, Novosibirsk State University
Hint: Consider the sum of squares of the roots of $q_n(x)$.