Problem

Source: All-Russian MO 2009 Regional 10. 8

Tags: combinatorics



At a party, a group of $20$ people needs to be seated at $4$ tables. The seating arrangement is called successful if any two people at the same table are friends. It turned out that successful seating arrangements exist. In a successful seating arrangement, exactly $5$ people sit at each table. What is the greatest possible number of pairs of friends in this companies?