maths by Matija

A group of mathematicians is attending a conference. We say that a mathematician is $k$-content if he is in a 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 a 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.