Question 17
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 $k$?