# Task 6. (14 points)

According to the regulations of the chess tournament, each participant must play one game with every other participant. After exactly 99 games were played, it turned out that the set of participants could be divided into two unequal groups such that all opponents belonging to the same group had already played their games with each other. At the same time, no more than four games were played between opponents belonging to different groups. What is the maximum possible number of participants in this chess tournament?
# Solution.

Let the number of participants in the tournament be $n$, and the number of those who fell into the 1st group be $k$. Then the number of matches played is:

$\frac{k(k-1)}{2}+\frac{(n-k)((n-k)-1)}{2}+m=99$, where $m$ is small, $\Rightarrow$

$k^{2}-k+(n-k)^{2}-(n-k)+2 m-198=0$

$k^{2}-k+k^{2}-2 k n+n^{2}-n+k+2 m-198=0$

$2 k^{2}-2 k n+\left(n^{2}-n+2 m-198\right)=0$

$\frac{D}{4}=n^{2}-2\left(n^{2}-n+2 m-198\right)=-n^{2}+2 n-4 m+396 \geq 0$

$n^{2}-2 n+(4 m-396) \leq 0 \quad(n-1)^{2}-(397-4 m) \leq 0$

$n_{\max }=1+\sqrt{397-4 m} \leq 20$.

If $n=20$, then

$2 k^{2}-40 k+(400-20+2 m-198)=0$

$k^{2}-20 k+91+m=0$

$k=10, n-k=10, m=9$ - contradiction;

$k=11, n-k=9, m=8$ - contradiction;

$k=12, n-k=8, m=5$ - contradiction;

$k=13, n-k=7, m=0$ - contradiction;

for $k>13 m13 m$ of the maximum number of participants in the tournament. There is no strict justification for the estimates provided or the answer. | $\pm$ | 11 |
| ![](https://cdn.mathpix.com/cropped/2024_05_06_c119ef58862eb5cdf0f6g-6.jpg?height=174&width=1357&top_left_y=427&top_left_x=204) | $+/ 2$ | 7 |
| The answer is correct. Correct but unexplained estimates for the maximum number of participants in the tournament are provided. A strict justification for the answer based on the given estimates is provided. |  |  |
| The answer is incorrect or missing. Justified estimates for the maximum number of participants in the tournament or correct relationships from which they are derived are provided in the solution. | $\mp$ | 3 |
| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| Maximum score |  | 14 |