MathDB

Prove that: (1) In the complex plane, each line (except for the real axis) that crosses the origin has at most one point

{z}

, satisfy

\frac {1+z^{23}}{z^{64}}\in\mathbb R.

(2) For any non-zero complex number

{a}

and any real number

\theta

, the equation

1+z^{23}+az^{64}=0

has roots in

S_{\theta}=\left\{ z\in\mathbb C\mid\operatorname{Re}(ze^{-i\theta })\geqslant |z|\cos\frac{\pi}{20}\right\}.

Proposed by Yijun Yao

Problem Statement