In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 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?

Err this wording sucks but it's late. Looking for symmetry, we may let the lowest ten players each get points from each other and also points from the other fifteen players, who get points from each other and also points from the lowest ten players. Let the lowest ten players be . If beats and ties then each has points. Let the other fifteen players be . If beats then each has points. Now, consider only relations between the two groups. Tie with , and let edges denote some defeating some . Then and . Also, . Connect to , giving each edges and edges to each thus far. Now for each of the remaining , connect to , giving each edges and edges total for all .

Summarizing, there is at least one possible construction:

• beats and ties
• beats
• beats
• beats
• all unmentioned relationships contain some beating some