RO-TST 2015 Problem 18

Let $n$ be an integer greater than $1$ , and let $p$ be a prime divisor of $n$ . A confederation consists of $p$ states, each of which has exactly $n$ airports. There are $p$ air companies operating interstate flights only such that every two airports in different states are joined by a direct (two-way) flight operated by one of these companies. Determine the maximal integer $N$ satisfying the following condition: In every such confederation it is possible to choose one of the $p$ air companies and $N$ of the $np$ airports such that one may travel (not necessarily directly) from any one of the $N$ chosen airports to any other such only by flights operated by the chosen air company.