In a chess tournament with 12 participants, each played exactly one game against each of the others. A win was worth 1 point, a draw - $1 / 2$, and a loss - 0. Vasya lost only one game, but came in last place, scoring the fewest points. Petya came in first place, scoring the most points. By how many points did Vasya fall behind Petya?
In the tournament, a total of 66 games were played, which means the same number of points were awarded. According to the conditions, Vasya lost one game, so out of the 10 participants, he either drew or won. Therefore, he scored no less than 5 points. Consequently, each of the others scored no less than 5.5 points, and all the chess players together scored no less than $11 \cdot 5.5 + 5 = 65.5$ points. This is only possible if Petya, who took first place, scored 6 points, Vasya scored 5 points, and the other participants each scored 5.5 points.

## Answer

By one.