Problem 11.3. In a football tournament, 15 teams participated, each playing against each other exactly once. For a win, 3 points were awarded, for a draw - 1 point, and for a loss - 0 points.

After the tournament ended, it turned out that some 6 teams scored at least $N$ points each. What is the largest integer value that $N$ can take?

![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-42.jpg?height=494&width=460&top_left_y=94&top_left_x=499)

Fig. 13: to the solution of problem 11.2

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# Answer: 34.

Solution. Let's call these 6 teams successful, and the remaining 9 teams unsuccessful. We will call a game between two successful teams an internal game, and a game between a successful and an unsuccessful team an external game.

First, note that for each game, the participating teams collectively earn no more than 3 points. Clearly, there were exactly $\frac{6 \cdot 5}{2}=15$ internal games, and only from these games did all successful teams collectively earn no more than $15 \cdot 3=45$ points. There were exactly $6 \cdot 9=54$ external games, and in each such game, a successful team earned no more than 3 points. Therefore, collectively, all successful teams earned no more than $54 \cdot 3=162$ points from external games. According to the problem, the successful teams collectively earned at least $6 \mathrm{~N}$ points, so we get the inequality $6 N \leqslant 45+162$. From this, it follows that $N \leqslant \frac{207}{6}<35$ and $N \leqslant 34$. Now let's provide an example for $N=34$. Number the teams from 1 to 15. We will show how teams from 1 to 6 can earn at least 34 points.

- Suppose each team from 1 to 6 won against each team from 7 to 15, then each team from 1 to 6 earned $9 \cdot 3=27$ points just from these games.
- Suppose the teams from 1 to 6 played among themselves as indicated in the following table (in each cell, the number of points is shown that the team from the corresponding row received in the game with the team from the corresponding column):

|  | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{3}$ | $\mathbf{4}$ | $\mathbf{5}$ | $\mathbf{6}$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $\mathbf{1}$ |  | 3 | 3 | 1 | 0 | 0 |
| $\mathbf{2}$ | 0 |  | 3 | 3 | 1 | 0 |
| $\mathbf{3}$ | 0 | 0 |  | 3 | 3 | 1 |
| $\mathbf{4}$ | 1 | 0 | 0 |  | 3 | 3 |
| $\mathbf{5}$ | 3 | 1 | 0 | 0 |  | 3 |
| $\mathbf{6}$ | 3 | 3 | 1 | 0 | 0 |  |

- Suppose in each game between teams from 7 to 15, the team with the higher number won (the outcome of these games does not matter).

In total, teams from 1 to 6 earned at least $27+7=34$ points.