CMO 2025 Problem 3

A polynomial $c_{d}x^{d} + c_{d - 1}x^{d - 1} + \dots + c_{1}x + c_{0}$ with degree $d$ is reflexive if there is an integer $n \geq d$ such that $c_{i} = c_{n - i}$ for every $0 \leq i \leq n$ , where $c_{i} = 0$ for $i > d$ . Let $\ell \geq 2$ be an integer and $p(x)$ be a polynomial with integer coefficients. Prove that there exist reflexive polynomials $q(x), r(x)$ with integer coefficients such that

$$(1 + x + x^{2} + \dots + x^{\ell -1})p(x) = q(x) + x^{\ell}r(x).$$