KURSCHAK 2018 Problem 3

In a village (where only dwarfs live) there are $k$ streets, and there are $k(n-1)+1$ clubs each containing $n$ dwarfs. A dwarf can be in more than one clubs, and two dwarfs know each other if they live in the same street or they are in the same club (there is a club they are both in).

Prove that is it possible to choose $n$ different dwarfs from $n$ different clubs (one dwarf from each club), such that the $n$ dwarfs know each other!