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max of complex numbers' sum |z_i-z_j|^2 if |z_i-z_j| <=1

Source: 6th XMO (China) p2 https://artofproblemsolving.com/community/c3192772_geometry_regional_china

April 7, 2024
complex numbersalgebrainequalitiesgeometric inequality

Problem Statement

Assume that complex numbers z1,z2,...,znz_1,z_2,...,z_n satisfy ∣zi−zj∣≤1|z_i-z_j| \le 1 for any 1≤i<j≤n1 \le i <j \le n. Let S=∑1≤i<j≤n∣zi−zj∣2.S= \sum_{1 \le i <j \le n} |z_i-z_j|^2. (1) If n=6063n = 6063, find the maximum value of SS. (2) If n=2021n= 2021, find the maximum value of SS.
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