TH-TMO 2021 Problem 10

Let $d\geq 13$ be an integer, and let $P(x) = a_dx^d + a_{d-1}x^{d-1} + \dots + a_1x+a_0$ be a polynomial of degree $d$ with complex coefficients such that $a_n = a_{d-n}$ for all $n\in\{0,1,\dots,d\}$ . Prove that if $P$ has no double roots, then $P$ has two distinct roots $z_1$ and $z_2$ such that $|z_1-z_2|<1$ .