8.5. Eleven of the best football teams each played one match against each other. It turned out that each team scored 1 goal in their first match, 2 goals in their second match, ..., 10 goals in their tenth match. What is the maximum number of matches that could have ended in a draw?
Solution: According to the condition, each team scored 1 goal in the first game. In the case of a draw, it also conceded 1 goal. Then for the other team, this match was also the first. Since the number of teams is odd, they cannot be paired. Therefore, at least one of the teams played a non-draw in their first match. The same is true for the second, third, ..., tenth matches. Since each match is counted twice, there were at least 5 non-draw matches, and then the number of draws was no more than $11 \cdot 10 / 2 - 5 = 50$.

Let's provide an example of a tournament with 50 draws. Number the teams from 1 to 11. Form 5 pairs of adjacent teams (leaving the 11th team without a pair). First, the teams in the pairs play against each other, and then the 11th team plays against the 1st (with a score of $1:2$). Again, we pair the teams, this time excluding the 1st, and let the teams in the pairs play again (resulting in 5 draws). Now arrange the teams in a circle with a step of 2, specifically in the order 1357911246810 (each team has two new neighbors). Applying the same match scheme, we get another 11 games, one of which is non-draw. Then apply this scheme for steps of 3, 4, and 5.

We can even write out a specific table:

| All play the first game, all draws | $12,34,56,78,910$ |
| :--- | :--- |
| 11 - first game, 1 - second game | 111 |
| All play the second game, all draws | $23,45,67,89,1011$ |
| All play the third game, all draws | $13,57,911,24,68$ |
| 10 - third game, 1 - fourth game | 101 |
| All play the fourth game, all draws | $35,79,112,46,810$ |
| All play the fifth game, all draws | $14,710,25,811,36$ |
| 9 - fifth game, 1 - sixth game | 91 |
| All play the sixth game, all draws | $47,102,58,113,69$ |
| All play the seventh game, all draws | $15,92,610,37,114$ |
| 8 - seventh game, 1 - eighth game | 81 |
| All play the eighth game, all draws | $59,26,103,711,48$ |
| All play the ninth game, all draws | $16,115,104,93,82$ |
| 7 - ninth game, 1 - tenth game | 71 |
| All play the tenth game, all draws | $611,510,49,38,27$ |

## Criteria:

Only the answer - 0 points.

Only the estimate that there are no more than $50-2$ points.

Only an example with 50 draws - 3 points.

There is an estimate and a correct idea for the example, but there are inaccuracies - 5 or 6 points depending on the size of the inaccuracies.