Difference between revisions of "2015 AMC 8 Problems/Problem 24"
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This gives <math>3N=48</math>, as desired. The answer is <math>\boxed{\textbf{(B)}~48}</math>. | This gives <math>3N=48</math>, as desired. The answer is <math>\boxed{\textbf{(B)}~48}</math>. | ||
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+ | ==See Also== | ||
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+ | {{AMC8 box|year=2015|num-b=24|after=Last Problem}} | ||
+ | {{MAA Notice}} |
Revision as of 17:10, 25 November 2015
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?
Contents
Solution 1
Note that the equation rewrites to .
Now remark that if is a solution to this equation, then so is . This is because Thus, we can now take an "edge case" solution and work upward until both conditions ( and ) are met.
We see by inspection that is a solution. By the above work, we can easily deduce that and are solutions. The last one is the intended answer (the next solution fails ) so our answer is .
Solution 2
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.
Solution 3
, 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 24 |
Followed by Last Problem | |
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All AJHSME/AMC 8 Problems and Solutions |
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