Problem:
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 points, and each of the two players earned 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 in games 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?
Solution:
Assume that a total of players participated in the tournament. We will obtain two expressions in : one by considering the total number of points gathered by all of the players, and one by considering the number of points gathered by the losers ( lowest scoring contestants) and those gathered by the winners (other contestants) separately. To obtain the desired expressions, we wil1 use the fact that if players played against one another, then they played a total of games, resulting in a total of points to be shared among them.
In view of the last observation, the players gathered a total of points in the tournament. Similarly, the losers had or 45 points in games among themselves; since this accounts for half of their points, they must have had a total of points. In games among themse1ves the winners similarly gathered points; this also accounts for only half of their total number of points (the other half coming from games against the losers), so their total was points. Thus we have the equation
which is equivalent to
Since the left member of this equation may be factored as , it follows that or . We discard the first of these in view of the following observation: if there were only players in the tournament, then there would have been only winners, and the total of their points would have been points, resulting in an average of points for each of them. This is less than the or points gathered, on the average, by each of the losers: Therefore, ; i.e., there were players in the tournament.
The problems on this page are the property of the MAA's American Mathematics Competitions