Difference between revisions of "2019 AMC 8 Problems/Problem 19"
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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 <math>3</math> points, so a team gets <math>3*2=6</math> points if they each win a game and lose a game. This case brings a total of <math>18+6=24</math> 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 <math>3</math> points, so a team gets <math>3*2=6</math> points if they each win a game and lose a game. This case brings a total of <math>18+6=24</math> points. | ||
| − | Therefore, we use Case 2 since it brings the greater amount of points, or <math>\boxed{24}</math> | + | Therefore, we use Case 2 since it brings the greater amount of points, or <math>\boxed{24}</math>, so the answer is <math>\boxed{C}</math> |
==See Also== | ==See Also== | ||
Revision as of 19:09, 20 November 2019
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 
See Also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
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