2015 AMC 12A Problem 13

Problem 13 of 25IntermediateNumber Theory

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?

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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.

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