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Difference between revisions of "2015 AMC 12A Problems/Problem 13"
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Therefore, answer choice <math>\boxed{\textbf{(E)}}</math> is false. If everyone ties, every team will be tied for the first place with 11 points each.
 
Therefore, answer choice <math>\boxed{\textbf{(E)}}</math> is false. If everyone ties, every team will be tied for the first place with 11 points each.
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~xHypotenuse
  
  
 
== See Also ==
 
== See Also ==
 
{{AMC12 box|year=2015|ab=A|num-b=12|num-a=14}}
 
{{AMC12 box|year=2015|ab=A|num-b=12|num-a=14}}

Revision as of 00:46, 23 April 2024

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?

$\textbf{(A)}\ \text{There must be an even number of odd scores.}\\ \qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\ \qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\ \qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\ \qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}$

Solution 1

We can eliminate answer choices $\textbf{(A)}$ and $\textbf{(B)}$ because there are an even number of scores, so if one is false, the other must be false too. Answer choice $\textbf{(C)}$ must be true since every team plays every other team, so it is impossible for two teams to lose every game. Answer choice $\textbf{(D)}$ must be true since each game gives out a total of two points, and there are $\binom{12}{2} = 66$ games, for a total of $132$ points. Answer choice $\boxed{\textbf{(E)}}$ is false. If everyone draws each of their 11 games, then every team will tie for first place with 11 points each.

  • Quick question, why does the answer key say 3?

    Solution 2

    We will proceed by process of elimination:


    $\textbf{(A)}$: We know that this must be true, since any tied results in a 1 point (which is odd) for both teams. Hence, there must be 0 or a positive even number of odd scores.

    $\textbf{(B)}$: This is true too, because each non-tie generates 2 points for the winner, and 0 points for the loser, which are both even scores. Hence, there must be 0 or a positive even number of even scores as well.

    $\textbf{(C)}$: This must be true since every team plays every other team, so it is impossible for two teams to lose every game.

    $\textbf{(D)}$: This is true as well, since any game gives out a net total of two points, and there are $\binom{12}{2} = 66$ games, for a total of $132$ points for any round-robin tournament.


    Therefore, answer choice $\boxed{\textbf{(E)}}$ is false. If everyone ties, every team will be tied for the first place with 11 points each.

    ~xHypotenuse


    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
    • Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2015_AMC_12A_Problems/Problem_13&oldid=218469"