International Mathematics Competition
for University Students
2017

IMC2017: Day 2, Problem 7

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 $q_n(x)$.

  Solution