NL-NWO 2011 Problem 3

In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives $3$ points and the losing team receives $0$ points. If the game is a draw, the two teams receive $1$ point each. Can the final scores of the six teams be six consecutive numbers $a,a +1,...,a + 5$ ? If so, determine all values of $a$ for which this is possible.