2003 AMC 10B Problem 15

Question

There are

100

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

28

players are given a bye, and the remaining

72

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

a prime number

divisible by

2

divisible by

5

divisible by

7

divisible by

11

Answer: E

Concepts:basic counting invariant

Difficulty rating: 1280

Small hint:

Each match eliminates exactly one player

Big hint:

All but the champion are eliminated, so count the number of eliminated players

Solution:

Each match eliminates exactly one player. Since

100

players start and all but the champion are eliminated, there are

99

matches.

Because

99=9 \cdot 11,

it is divisible by

11

but satisfies none of the other options.

Thus, the correct answer is E.

2003 AMC 10B Problem 15

Answer: E

Difficulty rating: 1280

Small hint:

Big hint:

Solution:

Problem 15 in other years