# Why aren't there 21 players in this tournament?

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?

Here is my approach: Let $n$ be the number of players. It is safe to assume that the player at first place has won all of his games, after all the number of games the first place player has won can vary and yet the number of players will stay the same. Then he has $n-1$ points and thus $\frac{n-1}{2} = 10$, so $n=21$.

Keep in mind that I am looking for the mistake in my reasoning not a solution to the problem itself.

• You can't assume the player in first place won all their games. There's no particular reason that this must be true, and in fact, we can see that it leads to a contradiction. In particular, suppose that the player in first place won all their games. Then, none of the other players gain any points by beating him - thus, it is as if the remaining $n-1$ players are playing a tournament identical to the original one - so we would be equally justified saying that player won the remaining $n-2$ games - half of which against the lowest $10$ players. That suggests the contradictory result $n=22$. Jun 27, 2015 at 14:21
• If my hint got you to the solution, it would be good to write it up for others. You can (I think after a delay) accept it if you want. Jun 27, 2015 at 14:33
• The solution can be found here : artofproblemsolving.com/wiki/index.php/1985_AIME_Problems/… Jun 27, 2015 at 14:45

You can't assume that the first place player has won all his games. What you can do is compute the total number of games played by $n$ players and recognize that that represents the total score. Similarly, how many points did the bottom ten players score playing among themselves?