Problems about a chess tournament

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TASKS
ABOUT CHESS
TOURNAMENT

1. 8 chess players took part in the tournament*). They all scored a different number of points, and the second prize-winner scored the same amount as all the chess players who took places from 5th to 8th. How did the chess players who took 3rd and 5th places play among themselves?

2. Two 7th grade students and several 8th grade students played in the tournament. The seventh-graders scored 8 points together, and all the eighth-graders scored equally. How many eighth-graders could participate in the tournament?

3. n chess players—grandmasters and masters—participated in the tournament. After the end of the tournament, it turned out that each student gained half of his points in the game against the masters. Prove that $\sqrt{n}$ is an integer.

4. A chess player plays at least one game a day for training and, in order not to overwork, no more than 12 a week. Prove that it is possible to find several consecutive days during which he plays exactly 20 games.

5. Every chess player in the tournament scored half of their points in matches with the participants who took the last three places. How many people took part in the tournament?

6. Prove that at the end of the tournament its participants can be re-determined in such a way that it turns out that not a single participant lost to the next one indefinitely after him.

*) We assume that chess players play one game at a time with each other, that is, the tournament is held in one round.