[Tournaments and tournament tables]

A school table tennis championship was held according to the Olympic system. The winner won six matches. How many participants in the tournament won more games than they lost? (In a tournament according to the Olympic system, participants are divided into pairs. Those who lost a game in the first round are eliminated. Those who won in the first round are paired again. Those who lost in the second round are eliminated, and so on. In each round, every participant found a pair.)
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. Therefore, 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 their losses. Since the number of participants halved after each round, 16 participants advanced to the third round.

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