Difference between revisions of "2003 AMC 10B Problems/Problem 15"

Latest revision as of 22:23, 7 September 2023

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

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

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

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

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

$\qquad\textbf{(E) } \text{divisible by 11}$

Solution

Solution 1

Notice that $99$ players need to be eliminated for there to be declared a winner. Notice also that every match eliminates exactly one person. Therefore, $99$ matches are needed to eliminate $99$ people and therefore declare a winner. The rest of the information is irrelevant. Therefore, the total number of matches is $\text{divisible by 11}, \boxed{\text{E}}$ .

Solution 2

28 people receive byes, so in the first round there are $36$ matches played. In the second round there are $28 + 36 = 64$ people, so there are 32 matches. In the subsequent rounds, there are $16, 8, 4, 2, 1$ matches played, for a total of $36 + 32 + 16 + 8 + 4 + 2 + 1 = 99$ matches. Therefore, the answer is $\rightarrow\textbf{(E) }\text{divisible by 11.}$

Video Solution by WhyMath

https://youtu.be/v-x_BO5aQDw

~savannahsolver

2003 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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@@ Line 18: / Line 18: @@
 ===Solution 2===
-people receive byes, so in the first round there are <math>36</math> matches played. In the second round there are <math>28 + 36 = 64</math> people, so there are 32 matches. In the subsequent rounds, there are <math>16, 8, 4, 2, 1</math> matches played, for a total of <math>36 + 32 + 16 + 8 + 4 + 2 + 1 = 99</math> matches. Divisible by 11 <math>\Rightarrow</math> <math>\boxed{(E)} </math>.
+people receive byes, so in the first round there are <math>36</math> matches played. In the second round there are <math>28 + 36 = 64</math> people, so there are 32 matches. In the subsequent rounds, there are <math>16, 8, 4, 2, 1</math> matches played, for a total of <math>36 + 32 + 16 + 8 + 4 + 2 + 1 = 99</math> matches. Therefore, the answer is <math>\rightarrow\textbf{(E) }\text{divisible by 11.}</math>
+==Video Solution by WhyMath==
+https://youtu.be/v-x_BO5aQDw
+~savannahsolver
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