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complex inequality

Source: Croatian MO 2004 4th Grade P1

April 9, 2021
InequalityComplex numbercomplex inequalityinequalities

Problem Statement

Let z1,…,znz_1,\ldots,z_n and w1,…,wnw_1,\ldots,w_n (n∈N)(n\in\mathbb N) be complex numbers such that ∣ϵ1z1+…+ϵnzn∣≤∣ϵ1w1+…+ϵnwn∣|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|holds for every choice of ϵ1,…,ϵn∈{−1,1}\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}. Prove that ∣z1∣2+…+∣zn∣2≤∣w1∣2+…+∣wn∣2.|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.
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