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0152 complex numbers inequalities 1st edition Round 5 p2

Source:

May 9, 2021
complex numbersinequalities1st editionalgebra

Problem Statement

Let mm be the greatest number such that for any set of complex numbers having the sum of all modulus of all the elements 11, there exists a subset having the modulus of the sum of the elements in the subset greater than mm. Prove that 14≤m≤12.\frac14 \le m \le \frac12.
(Optional Task for 3p) Find a smaller value for the RHS.
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