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Difference between revisions of "2012 AMC 10B Problems/Problem 15"

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==Problem==
 
==Problem==
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In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end on the tournament?
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In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
  
 
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 </math>
 
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 </math>
 +
 
==Solution 1==
 
==Solution 1==
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The total number of games in the tournament is 6*5/2= 15.  
+
The total number of games (and wins) in the tournament is <math>\frac{6 \times 5}{2}= 15</math>. A six-way tie is impossible as this would imply each team has 2.5 wins, so the maximum number of tied teams is five. Here's a chart of 15 games where five teams each have 3 wins:
−
Here's a poorly done chart of 15 games:
 
 
  |  1  2  3  4  5  6 |
 
  |  1  2  3  4  5  6 |
−
|1 X W  L W  L W|
+
|1 X W  L W  L W |
−
|2 L  X  W L W W|  
+
|2 L  X  W L W W |
−
|3 W L  X W  L W|
+
|3 W L  X W  L W |
−
|4 L W  L  X  W W|
+
|4 L W  L  X  W W |
−
|5 W L W  L  X W|
+
|5 W L W  L  X W |
−
|6 L   L  L  L  L  X|
+
|6 L L  L  L  L  X |
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The "X's" are for when it is where a team is set against itself, which cannot happen. The chart says that Team 6 has lost all of it's matches, which means that each of the other teams won against it. Then, alternating Wins and Losses were tried. It shows that it is possible for 5 teams to tie for the same amount of wins, which in this case is 3 wins.
+
The "X's" are for when it is where a team is set against itself, which cannot happen. The chart says that Team 6 has lost all of its matches, which means that each of the other teams won against it. Then, alternating Wins and Losses were tried. It shows that it is possible for 5 teams to tie for the same amount of wins, which in this case is 3 wins.
−
Thus, the answer is <math>\boxed{textbf{(D)}\5}</math>
+
Thus, the answer is <math>\boxed{\textbf{(D)}\ 5}</math>.
  
 
==Solution 2==
 
==Solution 2==
−
The total number of games in the tournament is 6*5/2= 15.
+
Note that the total number of matches is 15, and if 4 teams tie for the most wins then they can tie for 3 wins each, but the 5th team can also have 3 wins, so 4 + 1 = 5 is the maximum number of teams that could be tied for the most wins at the end of the tournament.
−
Now, we see which numbers from 1-6 divide 15.
 
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1, 3, and 5 divide 15. 5 is the largest of the 3 numbers, and it is possible for each of 5 teams to win 3 games.
 
−
Thus, the answer is <math>\boxed{textbf{(D)}\5}</math>
 
  
 
==See Also==
 
==See Also==

Latest revision as of 19:04, 13 July 2021

Problem

In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Solution 1

The total number of games (and wins) in the tournament is $\frac{6 \times 5}{2}= 15$. A six-way tie is impossible as this would imply each team has 2.5 wins, so the maximum number of tied teams is five. Here's a chart of 15 games where five teams each have 3 wins: 

|  1  2  3  4  5  6 |
|1 X  W  L  W  L  W |
|2 L  X  W  L  W  W |
|3 W  L  X  W  L  W |
|4 L  W  L  X  W  W |
|5 W  L  W  L  X  W |
|6 L  L  L  L  L  X |

The "X's" are for when it is where a team is set against itself, which cannot happen. The chart says that Team 6 has lost all of its matches, which means that each of the other teams won against it. Then, alternating Wins and Losses were tried. It shows that it is possible for 5 teams to tie for the same amount of wins, which in this case is 3 wins. Thus, the answer is $\boxed{\textbf{(D)}\ 5}$.

Solution 2

Note that the total number of matches is 15, and if 4 teams tie for the most wins then they can tie for 3 wins each, but the 5th team can also have 3 wins, so 4 + 1 = 5 is the maximum number of teams that could be tied for the most wins at the end of the tournament.

See Also

2012 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

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