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Romania NMO 2022 Grade 10 P3

Source: Romania National Olympiad 2022

April 21, 2022

complex numbersalgebraromania

Problem Statement

Let

Z\subset \mathbb{C}

be a set of

n

complex numbers,

n\geqslant 2.

Prove that for any positive integer

m

satisfying

m\leqslant n/2

there exists a subset

U

of

Z

with

m

elements such that

\Bigg|\sum_{z\in U}z\Bigg|\leqslant\Bigg|\sum_{z\in Z\setminus U}z\Bigg|.

Vasile Pop

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