Combinatorics
Source: ARO 2021 11.8
April 20, 2021
combinatorics
Problem Statement
Each girl among girls has balls; there are in total balls in colors, from each color there are balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has balls, colored in the distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has balls, colored in the colors.