2015 AMC 12A Problem 13
Attempt Problem 13 of the 2015 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12A solutions, or check the answer key.
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13.
A league with 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 points for every game it wins and point for every game it draws. Which of the following is not a true statement about the list of scores?
There must be an even number of odd scores.
There must be an even number of even scores.
There cannot be two scores of
The sum of the scores must be at least
The highest score must be at least
Answer: E
Small Hint:
Every game contributes exactly points in total, so the sum of all scores is fixed
Big Hint:
Consider what happens if every single game ends in a draw
Solution:
Each of the teams plays games, so games are played, and each game adds points to the list. The total of all scores is
If every game is a draw, each team scores so the highest score need not reach thus statement can fail. The other statements always hold: the sum the sum being even forces an even number of odd scores and hence an even number of even scores, and two teams cannot both score because their mutual game gives at least one of them a point.
Thus, the correct answer is E.
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