2019 IMO Problems/Problem 3
A social network has users, some pairs of whom are friends. Whenever user is friends with user , user is also friends with user . Events of the following kind may happen repeatedly, one at a time: Three users , , and such that is friends with both and , but and are not friends, change their friendship statuses such that and are now friends, but is no longer friends with , and no longer friends with . All other friendship statuses are unchanged. Initially, users have friends each, and users have friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.