100 boys and 100 girls
Source: 2022 Grosman Mathematical Olympiad P4
September 22, 2022
combinatoricsInequality
Problem Statement
Along a circle-shaped path are boys and girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other.
Prove that the sum of distances between pairs of the same gender is always less than or equal to the sum of distances between all pairs of a boy and a girl.