# 5. Variant 1.

The organizers of a ping-pong tournament have only one table. They call up two participants who have not yet played against each other. If, after the game, the losing participant has suffered their second defeat, they are eliminated from the tournament (there are no draws in tennis). After 29 games have been played, it turned out that all participants except two were eliminated. How many players participated in the tournament
Answer: 16.

Solution: Each player is eliminated after exactly two losses. In the situation where two "finalists" remain, the total number of losses is 29. If $n$ people have been eliminated from the tournament, they have collectively suffered $2n$ losses, while the "finalists" could have $0 (0+0)$, $1 (0+1)$, or $2 (1+1)$ losses. Considering that $2n$ and $2n+2$ are even numbers, we arrive at the equation $2n+1=29$, from which $n=14$, and the total number of participants is 16.