Difference between revisions of "2003 AMC 10B Problems/Problem 15"
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==Solution== | ==Solution== | ||
| − | Notice that <math> | + | It is irrelevant that the tournament started with <math>100</math> players. Technically, we are starting with <math>72</math>. Notice that <math>71</math> players need to be eliminated for there to be declared a winner. Also notice that every match eliminates exactly one person. Therefore, <math>71</math> matches are needed to eliminate <math>71</math> people and declare a winner. The total number of matches is <math>\boxed{\textbf{(A)}\ \text{a prime number.}}</math> |
==See Also== | ==See Also== | ||
{{AMC10 box|year=2003|ab=B|num-b=14|num-a=16}} | {{AMC10 box|year=2003|ab=B|num-b=14|num-a=16}} | ||
[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
Revision as of 19:26, 26 November 2011
Problem
There are 
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 
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
$\textbf{(A) } \text{a prime number}
\qquad\textbf{(B) } \text{divisible by 2}
\qquad\textbf{(C) } \text{divisible by 5}
\qquad\textbf{(D) } \text{divisible by 7}
\qquad\textbf{(E) } \text{divisible by 11}$ (Error compiling LaTeX. Unknown error_msg)
Solution
It is irrelevant that the tournament started with players. Technically, we are starting with . Notice that 
players need to be eliminated for there to be declared a winner. Also notice that every match eliminates exactly one person. Therefore, matches are needed to eliminate people and declare a winner. The total number of matches is 
See Also
| 2003 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
