Question 180
Let $C = \{z \in \mathbb{C} \mid |z| = 1\}$ be the unit circle in the complex plane. 240 complex numbers $z_1, z_2, \dots, z_{240} \in C$ (can be repeated) satisfy the following conditions:
(1) for any open arc $\Gamma$ of length $\pi$ on $C$, there are at most 200 $j$'s $(1 \le j \le 240)$ such that $z_j \in \Gamma$;
(2) for any open arc $\gamma$ of length $\frac{\pi}{3}$ on $C$, there are at most 120 $j$'s $(1 \le j \le 240)$ such that $z_j \in \gamma$.
Find the maximum of $|z_1 + z_2 + \dots + z_{240}|$.
(1) for any open arc $\Gamma$ of length $\pi$ on $C$, there are at most 200 $j$'s $(1 \le j \le 240)$ such that $z_j \in \Gamma$;
(2) for any open arc $\gamma$ of length $\frac{\pi}{3}$ on $C$, there are at most 120 $j$'s $(1 \le j \le 240)$ such that $z_j \in \gamma$.
Find the maximum of $|z_1 + z_2 + \dots + z_{240}|$.