Question 64
A group of mathematicians is attending a conference. We say that a mathematician is $k$-content if he is in room with at least $k$ people he admires or if he is admired by at least $k$ other people in the room. It is known that when all participants are in the same room then they are all at least $3k+1$-content. Prove that you can assign everyone into one of 2 rooms in a way that everyone is at least $k$-content in his room and neither room is empty. *Admiration is not necessarily mutual and no one admires himself.*