# Problem 8. (5 points)

32 volleyball teams participate in a tournament according to the following scheme. In each round, all remaining teams are randomly paired; if the number of teams is odd, one team skips this round. In each pair, one team wins and the other loses, as there are no draws in volleyball. After three losses, a team is eliminated from the tournament. When all teams except one are eliminated, this team is declared the winner and the tournament ends.

What is the minimum number of rounds the tournament can last?
# Solution:

For all teams to be eliminated except one, they must suffer at least 93 losses, meaning at least 93 matches must be played.

We also note that after each round, the number of teams decreases by at most half, because no more than half of the teams lose. In particular, in the final round, only 2 teams could have participated, and they played one match. In the penultimate round, a maximum of 4 teams participated and played a maximum of 2 matches. One round before that, a maximum of 4 matches were played, and another round before that, a maximum of 8. In total, in the last 4 rounds, no more than 15 matches were played.

This means that at least 78 matches were played before that. In each of the remaining rounds, no more than 16 matches were played, so there were at least 5 such rounds. In total, we get at least $4+5=9$ rounds.

The teams could manage in 9 rounds. Indeed, in the first 4 rounds, the teams could pair up, and in each pair, they could inflict 2 losses on each other. After this, each team has one loss left, and we get a standard Olympic system of 5 rounds, in each of which half of the teams are eliminated.

Answer: 9