# 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?

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

## 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, there are 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