Problem

Source: Romania NMO - 2018

Tags: complex numbers



Let $n \in \mathbb{N}_{\geq 2}.$ Prove that for any complex numbers $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n,$ the following statements are equivalent: a) $\sum_{k=1}^n|z-a_k|^2 \leq \sum_{k=1}^n|z-b_k|^2, \: \forall z \in \mathbb{C}.$ b) $\sum_{k=1}^na_k=\sum_{k=1}^nb_k$ and $\sum_{k=1}^n|a_k|^2 \leq \sum_{k=1}^n|b_k|^2.$