6. In a checkers tournament, students from 10th and 11th grades participated. Each player played against every other player once. A player received 2 points for a win, 1 point for a draw, and 0 points for a loss. There were 10 times more 11th graders than 10th graders, and together they scored 4.5 times more points than all the 10th graders. How many points did the most successful 10th grader score?
Answer: 20.

Solution: 1. Let $a$ be the number of tenth graders who participated in the tournament, earning $b$ points. Then, $10a$ eleventh graders played, earning $4.5b$ points. In each match, 2 points are played for, and a total of $11a$ players play $\frac{11 a(11 a-1)}{2}$ matches. Therefore, from the condition of the problem, we have the equation $11 a(11 a-1)=5.5 b$, from which it follows that $b=2 a(11 a-1)$.