in a tournament each player played exactly one game against each of the other players. in each game the winner was awarded point, the loser got $0$ points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. after the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (in particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). what was the total number of players in the tournament?
