Question

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?

There must be an even number of odd scores.

There must be an even number of even scores.

There cannot be two scores of $0.$

The sum of the scores must be at least $100.$

The highest score must be at least $12.$

Answer: E

Concepts:parity invariant counterexample

Difficulty rating: 1660

Solution:

Each of the $12$ teams plays $11$ games, so $\dfrac{12\cdot 11}{2} = 66$ games are played, and each game adds $2$ points to the list. The total of all scores is $66\cdot 2 = 132.$

If every game is a draw, each team scores $11,$ so the highest score need not reach $12;$ thus statement $\text{(E)}$ can fail. The other statements always hold: the sum $132 \ge 100,$ 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 $0$ because their mutual game gives at least one of them a point.

Thus, the correct answer is E.

2015 AMC 12A Problem 13

Answer: E

Difficulty rating: 1660

Solution:

Problem 13 in Other Years