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Subsets and their centers of mass in a unit sphere

Source: MIPT Undergraduate Competition 2019 1.4 and 2.4

August 17, 2020
geometry3D geometryspheretopology

Problem Statement

Suppose that in a unit sphere in Euclidean space, there are 2m2m points x1,x2,...,x2m.x_1, x_2, ..., x_{2m}. Prove that it's possible to partition them into two sets of mm points in such a way that the centers of mass of these sets are at a distance of at most 2m\frac{2}{\sqrt{m}} from one another.
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