There are $2n+1$ people. For each $n$ people there is somebody who is friend with each of them. Prove there is a "know-them-all" person.

Say a group $M$ is good if everyone in $M$ knows everyone in $M$. Note that such group exist (say with $2$ people).

Take maximal good subgroup $M$. If size of this group $M$ is $k\leq n$ then there is somebody who knows them all. So we can add him to this group and we get new good group $M'$ which is bigger then $M$. A contradiction. So $M$ is of size $k\geq n+1$. Then in $M^C$ we have at most $n$ people, so there is somebody in $M$ who know everybody in $M^C$. But then this one knows everybody.