Difference between revisions of "2012 AMC 10B Problems/Problem 15"
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<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== | + | ==Solution 1== |
| − | The total | + | The total number of games in the tournament is 6*5/2= 15. |
| − | |||
Here's a poorly done chart of 15 games: | Here's a poorly done chart of 15 games: | ||
| 1 2 3 4 5 6 | | | 1 2 3 4 5 6 | | ||
| Line 14: | Line 13: | ||
|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| | ||
| − | 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 | + | 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. |
| + | Thus, the answer is <math>\boxed{textbf{(D)}\5}</math> | ||
| − | Thus, the answer is <math>\boxed{textbf{(D)}\5</math> | + | ==Solution 2== |
| + | The total number of games in the tournament is 6*5/2= 15. | ||
| + | Now, we see which numbers from 1-6 divide 15. | ||
| + | 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== | ||
Revision as of 15:46, 19 February 2014
Contents
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 on the tournament?
Solution 1
The total number of games in the tournament is 6*5/2= 15. Here's a poorly done chart of 15 games:
| 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 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. Thus, the answer is $\boxed{textbf{(D)}\5}$ (Error compiling LaTeX. Unknown error_msg)
Solution 2
The total number of games in the tournament is 6*5/2= 15. Now, we see which numbers from 1-6 divide 15. 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 $\boxed{textbf{(D)}\5}$ (Error compiling LaTeX. Unknown error_msg)
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
