2003 AMC 10B Problems/Problem 15

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Problem

There are $100$ players in a single 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

$\textbf{(A) } \text{a prime number}

\qquad\textbf{(B) } \text{divisible by 2}

\qquad\textbf{(C) } \text{divisible by 5}

\qquad\textbf{(D) } \text{divisible by 7}

\qquad\textbf{(E) } \text{divisible by 11}$ (Error compiling LaTeX. Unknown error_msg)

Solution

It is irrelevant that the tournament started with $100$ players. Technically, we are starting with $72$. Notice that $71$ players need to be eliminated for there to be declared a winner. Also notice that every match eliminates exactly one person. Therefore, $71$ matches are needed to eliminate $71$ people and declare a winner. The total number of matches is $\boxed{\textbf{(A)}\ \text{a prime number.}}$

See Also

2003 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AMC 10 Problems and Solutions