11.6. In a round-robin chess tournament (each chess player plays one game against each other), 20 chess players participated, 6 of whom were from Russia. It is known that Vladimir, who scored more points than anyone else, took first place. Levon from Armenia took second place, also outscoring each of the other 18 chess players. What is the maximum total number of points that the Russian chess players could have scored? (In chess, one point is awarded for a win, half a point for a draw, and no points for a loss.)
Solution: Let's provide an example showing that Russian chess players could collectively score 96 points. Suppose Vladimir won all his games except the one against Levon, which ended in a draw. Additionally, suppose Levon drew all his games with the other Russians and consistently won against non-Russians. Finally, suppose all other games between Russians ended in draws, and all Russians defeated all other non-Russians except Levon. Then, Vladimir scored $18 \cdot 1 + 1 \cdot 0.5 = 18.5$ points, Levon scored $13 \cdot 1 + 6 \cdot 0.5 = 16$ points, each of the other Russians scored $13 \cdot 1 + 5 \cdot 0.5 = 15.5$ points, and each of the other thirteen players scored no more than 12 points. In this case, the condition of the problem is satisfied, and the total points scored by all Russians is $18.5 + 5 \cdot 15.5 = 96$.

We will show by contradiction that the Russians cannot score more points. Suppose they collectively scored 96.5 points or more. In the 15 games among themselves, they collectively scored 15 points, and the remaining (at least 81.5) points were scored in games with players from other countries. There were $6 \cdot 14 = 84$ games between a Russian and a non-Russian, and a total of 84 points were at stake in these games. Therefore, all non-Russians, including Levon, scored no more than 2.5 points in games with Russians, and no one, including Levon, could score more than $13 + 2.5 = 15.5$ points in total. The five Russians he outperformed scored no more than 15 points each, and thus Vladimir must have scored at least $96.5 - 15 \cdot 5 = 21.5$ points. However, he played only 19 games and could not score more than 19 points. This is a contradiction.

Answer: 96 points.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| Proved that Russians cannot collectively score 96.5 points or more, but no example of 96 points is provided | 4 points |
| Provided a correct example where Russians collectively scored 96 points; however, it is not proved that they could not score more points. | 2 points |
| Correct answer without justification (or with incorrect justification) | 1 point |
| Incorrect answer and/or reasoning that did not lead to the answer or precise estimate | 0 points |