Problem:
There are players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest players are given a bye, and the remaining players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is
Answer Choices:
A. a prime number
B. divisible by
C. divisible by
D. divisible by
E. divisible by
Solution:
In the first round players are eliminated, one per match. In the second round there are matches, in the third , then , and . The total number of matches is:
Note that is , but does not satisfy any of the other conditions given as answer choices.
In each match, precisely one player is eliminated. Since there were players in the tournament and all but one is eliminated, there must be matches.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions