CN-TST 2022 Problem 18

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of $z^{20}+63z+22=0$ is greater than $\pi$ . (2) Let $a_1,a_2,\ldots,a_n$ be complex numbers with sum $1$ , and $k_1<k_2<\cdots<k_n$ be odd positive integers. Let $\omega$ be a complex number with norm at least $1$ . Prove that the equation $$a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w$$ has at least one complex root with norm at most $3n|\omega|$ .