2019 AMC 8 Problems/Problem 19
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.
Solution 3
We can name the top three teams as and . We can see that , because these teams have the same points. If we look at the matches that involve the top three teams, we see that there are some duplicates: and come twice. In order to even out the scores and get the maximum score, we can say that in match and each win once out of the two games that they play. We can say the same thing for and . This tells us that each team and win and lose twice. This gives each team a total of 3 + 3 + 0 + 0 = 6 points. Now, we need to include the other three teams. We can label these teams as and . We can write down every match that or plays in that we haven't counted yet: and . We can say and win each of these in order to obtain the maximum score that and can have. If and win all six of their matches, and will have a score of . results in a maximum score of . This tells us that the correct answer choice is .
~Champion1234
Video Solutions
Associated Video - https://youtu.be/s0O3_uXZrOI
Video Solution - https://youtu.be/Lw8fSbX_8FU (Also explains problems 11-20)
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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