2019 AMC 8 Problems/Problem 19
Problem 19
In a tournament there are six team that play each other twice. A team earns points for a win,
point for a draw, and
points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?
Solution 1
After fully understanding the problem, we immediately know that the three top teams, say team , team
, and team
, must beat the other three teams
,
,
. Therefore,
,
,
must each obtain
points. However, they play against each team twice, for a total of
points against
,
, and
. For games between
,
,
, we have 2 cases. In both cases, there is an equality of points between
,
, and
.
Case 1: A team ties the two other teams. For a tie, we have 1 point, so we have points (they play twice). Therefore, this case brings a total of
points.
Case 2: A team beats one team while losing to another. This gives equality, as each team wins once and loses once as well. For a win, we have points, so a team gets
points if they each win a game and lose a game. This case brings a total of
points.
Therefore, we use Case 2 since it brings the greater amount of points, or , so the answer is
.
~A1337h4x0r
Note that case 2 can be easily seen to be better as follows. Let be the number of points
gets,
be the number of points
gets, and
be the number of points
gets. Since
, to maximize
, we can just maximize
. But in each match, if one team wins then the total sum increases by
points, whereas if they tie, the total sum increases by
points. So it is best if there are the fewest ties possible.
See Also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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