Difference between revisions of "2015 AMC 8 Problems/Problem 24"
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On one team they play <math>\binom{3}{2}N</math> games in their division and <math>4(M)</math> games in the other. This gives <math>3N+4M=76</math> | On one team they play <math>\binom{3}{2}N</math> games in their division and <math>4(M)</math> games in the other. This gives <math>3N+4M=76</math> | ||
Revision as of 12:50, 29 May 2019
A baseball league consists of two four-team divisions. Each team plays every other team in its division games. Each team plays every team in the other division games with and . Each team plays a 76 game schedule. How many games does a team play within its own division?
Solution 1
On one team they play games in their division and games in the other. This gives
Since we start by trying . This doesn't work because is not divisible by .
Next, does not work because is not divisible by
We try this does work giving and thus games in their division.
seems to work,until we realize this gives , but so this will not work.
Solution 2
, giving . Since , we have . Since is , we must have equal to , so .
This gives , as desired. The answer is .
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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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