2015 AMC 12A Problems/Problem 13

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

We can eliminate answer choices and because there are an even number of scores, so if one is false, the other must be false too. Answer choice must be true since every team plays every other team, so it is impossible for two teams to lose every game. Answer choice must be true since each game gives out a total of two points, and there are games, for a total of points. 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.