2015 AMC 12A Problem 13

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

Small hint:

Every game contributes exactly

2

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

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.