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?
Answer Choices:
A. There must be an even number of odd scores.
B. There must be an even number of even scores.
C. There cannot be two scores of 0 .
D. The sum of the scores must be at least 100 .
E. The highest score must be at least 12.
Solution:
Note that each of the 12 teams plays 11 games, so games are played in all. If every game ends in a draw, then each team will have a score of 11, so statement is not true. Each of the other statements is true. Each of the games generates 2 points in the score list, regardless of its outcome, so the sum of the scores must be ; thus (D) is true. Because the sum of an odd number of odd numbers plus any number of even numbers is odd, and 132 is even, there must be an even number of odd scores; thus (A) is true. Because there are 12 scores in all, there must also be an even number of even scores; thus (B) is true. Two teams cannot both have a score of 0 because the game between them must result in 1 point for each of them or 2 points for one of them; thus (C) is true.
The problems on this page are the property of the MAA's American Mathematics Competitions