MathDB
Inequality on the unit circle

Source: 2022 China TST, Test 1, P5

March 24, 2022
inequalitiescomplex numbers

Problem Statement

Let C={z∈C:∣z∣=1}C=\{ z \in \mathbb{C} : |z|=1 \} be the unit circle on the complex plane. Let z1,z2,…,z240∈Cz_1, z_2, \ldots, z_{240} \in C (not necessarily different) be 240240 complex numbers, satisfying the following two conditions: (1) For any open arc Γ\Gamma of length π\pi on CC, there are at most 200200 of j (1≤j≤240)j ~(1 \le j \le 240) such that zj∈Γz_j \in \Gamma. (2) For any open arc γ\gamma of length π/3\pi/3 on CC, there are at most 120120 of j (1≤j≤240)j ~(1 \le j \le 240) such that zj∈γz_j \in \gamma.
Find the maximum of ∣z1+z2+…+z240∣|z_1+z_2+\ldots+z_{240}|.
Back to ProblemsView on AoPS