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Romanian National Olympiad 2018 - Grade 10 - problem 3

Source: Romania NMO - 2018

April 12, 2018
complex numbers

Problem Statement

Let n∈N≥2.n \in \mathbb{N}_{\geq 2}. Prove that for any complex numbers a1,a2,…,ana_1,a_2,\ldots,a_n and b1,b2,…,bn,b_1,b_2,\ldots,b_n, the following statements are equivalent: a) ∑k=1n∣z−ak∣2≤∑k=1n∣z−bk∣2, ∀z∈C.\sum_{k=1}^n|z-a_k|^2 \leq \sum_{k=1}^n|z-b_k|^2, \: \forall z \in \mathbb{C}. b) ∑k=1nak=∑k=1nbk\sum_{k=1}^na_k=\sum_{k=1}^nb_k and ∑k=1n∣ak∣2≤∑k=1n∣bk∣2.\sum_{k=1}^n|a_k|^2 \leq \sum_{k=1}^n|b_k|^2.
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