MathDB

Drawing disjoint arcs on the unit sphere

Source: 42nd International Tournament of Towns, Senior A-Level P7, Spring 2021

February 18, 2023

combinatoricscombinatorial geometryTournament of Towns

Problem Statement

An integer

n > 2

is given. Peter wants to draw

n{}

arcs of length

\alpha{}

of great circles on a unit sphere so that they do not intersect each other. Prove that
[*]for all

\alpha<\pi+2\pi/n

it is possible; [*]for all

\alpha>\pi+2\pi/n

it is impossible;
Ilya Bogdanov