10. $n>10$ teams take part in a soccer tournament. Each team plays every other team exactly once. A win gives two points, a draw one point and a defeat no points. After the tournament, it turns out that each team has won exactly half of its points in the games against the 10 worst teams (in particular, each of these 10 teams has scored half of its points against the 9 remaining teams). Determine all possible values of $n$, and give an example of such a tournament for these values.

## 1st solution

We call the 10 worst teams the losers, the $n-10$ best teams the winners. We repeatedly use the following fact: If $k$ teams play against each other, then the total number of points won is exactly $k(k-1)$. We count the total number of points won in two ways. On the one hand, this is exactly $n(n-1)$. On the other hand, the 10 losers in the games among themselves receive exactly $10 \cdot 9=90$ points. By assumption, this is exactly half of the total number of points that these 10 teams have achieved. Consequently, the total number of points scored by the losers is 180. The $n-10$ winners scored a total of $(n-10)(n-11)$ points in the games among themselves. Again, this is half of the total number of points, the latter is therefore equal to $2(n-10)(n-11)$. A comparison yields the equation

$$
n(n-1)=180+2(n-10)(n-11)
$$

This is equivalent to $n^{2}-41 n+400=0$ and has the solutions $n=16$ and $n=25$.

According to the above calculations, it also follows that the average score of the $n-10$ winners is equal to 2( $n-11)$, the average score of the 10 losers is equal to 18.

Of course, $2(n-10) \geq 18$ must now apply, i.e. $n \geq 20$. Consequently, $n=16$ is not possible.

Finally, we show that such a tournament exists for $n=25$. The 10 losers always play to a draw among themselves, as do the $n-10$ winners among themselves. The following diagram shows the games of the 15 winners $G_{i}$ against the 10 losers $V_{j}$. Where 2 means a win for $G_{i}$, 0$ a win for $V_{j}$ and 1 a draw.

| | $G_{1}$ | $G_{2}$ | $G_{3}$ | $G_{4}$ | $G_{5}$ | $G_{6}$ | $G_{7}$ | $G_{8}$ | $G_{9}$ | $G_{10}$ | $G_{11}$ | $G_{12}$ | $G_{13}$ | $G_{14}$ | $G_{15}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $V_{1}$ | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 0 |
| $V_{2}$ | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 2 | 2 | 2 | 0 |
| $V_{3}$ | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 2 | 2 | 2 |
| $V_{4}$ | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 1 | 2 | 2 | 2 |
| $V_{5}$ | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 1 | 2 | 2 |
| $V_{6}$ | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 0 | 1 | 2 | 2 |
| $V_{7}$ | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 0 | 1 | 2 |
| $V_{8}$ | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 0 | 1 | 2 |
| $V_{9}$ | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 0 | 1 |
| $V_{10}$ | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 0 | 1 |

In total, the final score of each winner is 28, of which exactly 14 points come from games against the losers. Each loser has a final score of 18, half of which again comes from games against the other losers. This shows everything.

## 2nd solution

Here is another example of such a tournament. Again, the losers and the winners play each other to a draw. Now assume that a winner never loses to a loser. A winner scores a total of 14 points against the other winners, so he must also score 14 points against the losers. To do this, he must win four times and draw six times. Similarly, each loser draws nine times. We can now divide the $10 \times 15$ table from above into $2 \times 3$ boxes and obtain the following table with the same notations:

| | $G_{1}, G_{2}, G_{3}$ | $G_{4}, G_{5}, G_{6}$ | $G_{7}, G_{8}, G_{9}$ | $G_{10}, G_{11}, G_{12}$ | $G_{13}, G_{14}, G_{15}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $V_{1}, V_{2}$ | 1 | 1 | 1 | 2 | 2 |
| $V_{3}, V_{4}$ | 2 | 1 | 1 | 1 | 2 |
| $V_{5}, V_{6}$ | 2 | 2 | 1 | 1 | 1 | 1 |
| $V_{7}, V_{8}$ | 1 | 2 | 2 | 1 | 1 |
| $V_{9}, V_{10}$ | 1 | 1 | 2 | 2 | 1 |
No matching solution found.