VN-TST 1992 Problem 6

In a scientific conference, all participants can speak in total $2 \cdot n$ languages ( $n \geq 2$ ). Each participant can speak exactly two languages and each pair of two participants can have at most one common language. It is known that for every integer $k$ , $1 \leq k \leq n-1$ there are at most $k-1$ languages such that each of these languages is spoken by at most $k$ participants. Show that we can choose a group from $2 \cdot n$ participants which in total can speak $2 \cdot n$ languages and each language is spoken by exactly two participants from this group.