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?
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
.