Difference between revisions of "2015 AMC 12A Problems/Problem 13"
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A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores? | A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores? | ||
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==Solution== | ==Solution== | ||
Answer choice <math>\textbf{(E)}</math> is false (and thus our answer). If everyone draws each of their 11 games, then every team will tie for first place with 11 points each. | Answer choice <math>\textbf{(E)}</math> is false (and thus our answer). If everyone draws each of their 11 games, then every team will tie for first place with 11 points each. | ||
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| + | == See Also == | ||
| + | {{AMC12 box|year=2015|ab=A|num-b=12|num-a=14}} | ||
Revision as of 00:50, 5 February 2015
Problem
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?
$\textbf{(A)}\ \text{There must be an even number of odd scores.}\\ \qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\ \qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\ \qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\ \qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}$ (Error compiling LaTeX. Unknown error_msg)
Solution
Answer choice 
is false (and thus our answer). If everyone draws each of their 11 games, then every team will tie for first place with 11 points each.
See Also
| 2015 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 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 12 Problems and Solutions | |
