2015 AMC 8 Problems/Problem 24
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?
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 .
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.