2. The school table tennis championship was held according to the Olympic system. The winner won 6 matches. How many participants in the championship won more matches than they lost? (In the first round of the championship held according to the Olympic system, participants are paired. Those who lost the first game are eliminated from the championship, while those who won in the first round are paired again to play the second round. The losers are eliminated again, the winners are paired for the third round, and so on, until only one champion remains. It is known that in each round of our championship, a pair was found for each participant.)
Answer: 16. Solution. Since in each round every player found a partner and in each pair one of the players was eliminated, the total number of players decreased by half after each round. The winner participated in every round and won, so there were a total of six rounds. Since the winner was determined unequivocally after the sixth round, the total number of participants was $2^{6}=64$. Those who lost in the first round have one loss and zero wins, those who lost in the second round have one win and one loss. All those who advanced to the third round will have at least two wins and no more than one loss (after which they were eliminated), meaning the number of their wins is greater than the number of losses. Since the number of participants decreased by half after each round, 16 participants advanced to the third round.

Grading Instructions. The solution relies on two facts: (1) there were a total of 64 participants in the tournament; (2) the number of participants decreased exactly by half after each round. For the correct answer without any explanation - 1 point. For the correct answer with reasoning, explicitly or implicitly relying on facts (1) and (2), without justifying both of these facts - no more than 3 points. Without justifying fact (1) - no more than 5 points. Without justifying fact (2) - no more than 6 points.