2. At the international table tennis championship, 200 participants gathered. The game is played in a knockout format, i.e., in each match, two players participate, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who won at least three matches.
Answer: 66.

Solution: Each participant (except the winner) lost one game to someone. There are 199 such participants, so no more than 66 participants could have won 3 games (someone must lose 3 games to them).

We will show that there could be 66 such participants. Let №4 win against №1,2,3; №7 - against №4,5,6,... №199 - against №196,197,198, and №200 win against №199. Then all participants with numbers giving a remainder of 1 when divided by 3 (except the first) won exactly 3 games.