Question 3
A social network has $2019$ users, some pairs of which are friends (friendship is symmetric). If $A$, $B$, $C$ are three users such that $AB$ are friends and $AC$ are friends but $BC$ is not, then the administrator may perform the following operation: change the friendships such that $BC$ are friends, but $AB$ and $AC$ are no longer friends.
Initially, $1009$ users have $1010$ friends and $1010$ users have $1009$ friends. Prove that the administrator can make a sequence of operations such that all users have at most $1$ friend.
Initially, $1009$ users have $1010$ friends and $1010$ users have $1009$ friends. Prove that the administrator can make a sequence of operations such that all users have at most $1$ friend.