Question 178
(1) Prove: on the complex plane, for each line passing through the origin except for the real axis, there is at most one point $z$ such that $\frac{1+z^{23}}{z^{64}}$ is a real number.
(2) Prove: for any complex number $a \ne 0$ and any real number $\theta$, the equation
$$
1 + z^{23} + a z^{64} = 0
$$
has at least one root in the set
$$
S_{\theta} = \{z \in \mathbb{C} \mid \mathrm{Re}(z \cdot e^{-i\theta}) \ge |z| \cdot \cos \frac{\pi}{20}\}.
$$
(2) Prove: for any complex number $a \ne 0$ and any real number $\theta$, the equation
$$
1 + z^{23} + a z^{64} = 0
$$
has at least one root in the set
$$
S_{\theta} = \{z \in \mathbb{C} \mid \mathrm{Re}(z \cdot e^{-i\theta}) \ge |z| \cdot \cos \frac{\pi}{20}\}.
$$