4

Part of 2018 Spain Mathematical Olympiad

Problems(1)

Coloring points on a spherical surface.

Source: Spain Mathematical Olympiad 2018 P4

Points on a spherical surface with radius

4

4

different colors. Prove that there exist two points with the same color such that the distance between them is either

4\sqrt{3}

2\sqrt{6}

.
(Distance is Euclidean, that is, the length of the straight segment between the points)

geometrySpaincombinatoricspigeonhole principle