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?
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 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.