RU-ALLRUSSIAN 1975 Problem 14

20 teams each play one game with every other team. Each game results in a win or loss (no draws). k of the teams are European. A separate trophy is awarded for the best European team on the basis of the k(k- 1)/2 games in which both teams are European. This trophy is won by a single team. The same team comes last in the overall competition (winning fewer games than any other team). What is the largest possible value of k? If draws are allowed and a team scores 2 for a win and 1 for a draw, what is the largest possible value of k?