In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in case of a tie, the order is arbitrary). It is then noticed that each participant has won half of their points against the bottom ten in the ranking. How many people participated in the tournament
We proceed by double-counting on the number of points distributed in the tournament. Each match corresponds to the distribution of 1 point, and thus the total score obtained by $k$ in the matches they play against each other is $k(k-1) / 2$. Therefore, the total score of the participants is $n(n-1) / 2$.

On the other hand, the last 10 participants scored 45 points by playing against each other. Their total score in the tournament is therefore 90 points. The other $n-10$ participants scored $(n-10)(n-11) / 2$ points by playing against each other, and thus a total of $(n-10)(n-11)$ points. The total score of all participants is therefore $90+(n-10)(n-11)$ points. By comparing these two values, we obtain the equation $\mathfrak{n}^{2}-41 \mathfrak{n}+400=0$, which has two solutions: 16 and 25.

The tournament could not have contained 16 participants. Indeed, in this case, the total score of the last 10 participants is 90, and that of the first 6 is 30. The last 10 participants would therefore have scored on average more points than the first, which is absurd. The tournament could therefore only have had 25 participants.