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 $|z_1-z_2|<1$.
Note $P(x)=x^dP(\frac 1x)$, so if $x$ is a root of $P$, then so is $\frac 1x$.
Now we can show that there are at least 7 roots inside (including the boundary of) the unit circle, so we are done by splitting the unit circle into 6 equal 60 degree sectors (an edge belongs to exactly one sector, not including the origin; if one point is on the origin, then 0 is a root, so $a_0=0$, contradiction. ), noting two points in the same sector has distance less than 1 by cosine law. By PHP, two points must be in the same sector, so we are done.