Problem 3. In a chess tournament, each of the participants played one game with each of the other participants. At the end of the tournament, the organizer also played one game with a few of the participants, so in total 100 games were played.

What was the number of participants and how many games did the tournament organizer play?
Solution. If the number of participants is $n$, then the number of games played by

![](https://cdn.mathpix.com/cropped/2024_06_07_3dd00a25927e70aa5fc0g-2.jpg?height=49&width=1589&top_left_y=756&top_left_x=230)

The total number of games is strictly less than 100, so $(n-1) \cdot n < 200$, which implies $n \leq 14$

$2 p$

Assuming $n \leq 13$, the number of games played among the participants is at most equal to $12 \cdot 13 / 2 = 78$, then the number of games played by the organizer would be at least equal to 22, which is greater than the number of participants, contradicting the problem's hypothesis

$2 p$

For $n=14$, the participants in the tournament play 91 games among themselves, and the organizer plays 9 games