Omni-MATH / Let C={z∈C:|z|=1} be the unit circle on the complex plane. Let z₁,z₂,…,z₂₄₀∈C (not necessarily different) be 240 complex numbers, satisfying the following two conditions: (1) For a…

Let C={z∈C:|z|=1} be the unit circle on the complex plane. Let z₁,z₂,…,z₂₄₀∈C (not necessarily different) be 240 complex numbers, satisfying the following two conditions:
(1) For any open arc Γ of length π on C, there are at most 200 of $j ~(1 \le j \le 240)$ such that z_(j)∈Γ.
(2) For any open arc γ of length π/3 on C, there are at most 120 of  $j ~(1 \le j \le 240)$ such that z_(j)∈γ.

Find the maximum of |z₁+z₂+…+z₂₄₀|.

Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions:
(1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ such that $z_j \in \Gamma$.
(2) For any open arc $\gamma$ of length $\pi/3$ on $C$, there are at most $120$ of  $j ~(1 \le j \le 240)$ such that $z_j \in \gamma$.

Find the maximum of $|z_1+z_2+\ldots+z_{240}|$.