Question 193
(1) Prove that on the complex plane, the convex hull of all complex roots of the equation
$$
z^{20} + 63z + 22 = 0
$$
is larger than $\pi$.
(2) Let $n$ be a positive integer, and $1 \le k_1 \lt k_2 \lt \dots \lt k_n$ be $n$ odd integers. Prove that for any $n$ complex numbers $a_1, a_2, \dots, a_n$ with sum 1 and any complex number $w$ with absolute value at least 1, the equation
$$
a_1z^{k_1} + a_2z^{k_2} + \dots + a_nz^{k_n} = w
$$
has at least one complex root with norm less than or equal to $3n|w|$.
$$
z^{20} + 63z + 22 = 0
$$
is larger than $\pi$.
(2) Let $n$ be a positive integer, and $1 \le k_1 \lt k_2 \lt \dots \lt k_n$ be $n$ odd integers. Prove that for any $n$ complex numbers $a_1, a_2, \dots, a_n$ with sum 1 and any complex number $w$ with absolute value at least 1, the equation
$$
a_1z^{k_1} + a_2z^{k_2} + \dots + a_nz^{k_n} = w
$$
has at least one complex root with norm less than or equal to $3n|w|$.