Problem #1066

1066.

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 tournament continues until only one player remains unbeaten. The total number of matches played is

$\textbf{(A) } \text{a prime number} \qquad\textbf{(B) } \text{divisible by 2} \qquad\textbf{(C) } \text{divisible by 5} \qquad\textbf{(D) } \text{divisible by 7} \qquad\textbf{(E) } \text{divisible by 11}$

This problem is copyrighted by the American Mathematics Competitions.

Note: you aren't logged in. If you log in, we'll keep a record of which problems you've solved.

Instructions for entering answers:

  • Reduce fractions to lowest terms and enter in the form 7/9.
  • Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
  • Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
  • Exponents should be entered in the form 10^10.
  • If the problem is multiple choice, enter the appropriate (capital) letter.
  • Enter points with parentheses, like so: (4,5)
  • Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

For questions or comments, please email markan@eudelic.com.

Registration open for AMC10/12 prep class

Registration is now open. See details here.