Solution
= Solution
Define the continuous map $T(z)=-(z^6-z^5+2z^2+1)/6$. For $|z|\le1$, the <triangle inequality> gives $|T(z)|\le(1+1+2+1)/6=5/6$. Thus $T$ maps the closed unit disk into itself. By the <Brouwer fixed-point theorem>, some $z$ has $T(z)=z$, which rearranges to
$$
\boxed{z^6-z^5+2z^2+6z+1=0,\qquad |z|\le5/6.}
$$
This supplies the requested existence and even places a root strictly inside the unit disk.