Problem

Source: Romanian TST 2 2007, Problem 1

Tags: algebra, polynomial, algebra proposed



Let \[f = X^{n}+a_{n-1}X^{n-1}+\ldots+a_{1}X+a_{0}\] be an integer polynomial of degree $n \geq 3$ such that $a_{k}+a_{n-k}$ is even for all $k \in \overline{1,n-1}$ and $a_{0}$ is even. Suppose that $f = gh$, where $g,h$ are integer polynomials and $\deg g \leq \deg h$ and all the coefficients of $h$ are odd. Prove that $f$ has an integer root.