Task 9. The Federation of Wrestling has assigned each participant in the competition a qualification number. It is known that in matches between wrestlers whose qualification numbers differ by more than 2, the wrestler with the lower number always wins. The tournament for 256 wrestlers is held on an Olympic system: at the beginning of each day, the wrestlers are paired, the loser is eliminated from the competition (there are no draws). What is the highest qualification number that the winner can have?
Answer: 16

Solution. Note that a wrestler with number $k$ can only lose to a wrestler with number $k+1$ or $k+2$, so after each round, the smallest number cannot increase by more than 2 numbers. In a tournament with 256 participants, there are 8 rounds, so the number of the tournament winner does not exceed $1+2 \cdot 8=17$.

Suppose the wrestler with number 17 can win. Then in the first round, the wrestlers with numbers 1 and 2 must be eliminated. This is only possible if the wrestler with number 1 loses to the wrestler with number 3, and the wrestler with number 2 loses to the wrestler with number 4. Thus, after the first round, the wrestlers with numbers 3 and 4 will remain. Similarly, after the second round, the wrestlers with numbers 5 and 6 will remain, after the third round - 7 and $8, \ldots$, after the seventh round - 15 and 16. This means that in the final, decisive match, the wrestlers with numbers 15 and 16 will meet. Contradiction with the assumption that the wrestler with number 17 can win.

We will show that the wrestler with number 16 can win. Let's call the wrestlers with numbers greater than 16 weak. Suppose in the round with number $k \leqslant 7$ the wrestler with number $2 k-1$ loses to the wrestler with number $2 k+1$, the wrestler with number $2 k$ loses to the wrestler with number $2 k+2$, the wrestlers with numbers $2 k+3, \ldots, 16$ win against some weak wrestlers, and the remaining weak wrestlers play among themselves. Then after 7 rounds, the wrestlers with numbers 15 and 16 will remain, and in the final match, the wrestler with number 16 will win.