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and 
be positive integers, and let 
be a 
grid of unit squares.
or 
rectangle. A subset 
of grid squares in is domino-tileable if dominoes can be placed to cover every square of exactly once with no domino extending outside of . Note: The empty set is domino tileable. to the upper-right corner of formed by exactly 
edges of the grid squares. and , the number of up-right paths that divide into two domino-tileable subsets.
times as many students as Lara’s high school. The two high schools have a total of
students. How many students does Portia’s high school have?
and 
lie in a straight line in that order. Suppose that 
is a point not on the line and that 
and 
Find the area of 
to 
is 24 from herons on 
Gives 
Then, we know 
so 
therefore 
so 
By Heron's we get the area of 
has the same height as 
so the area of 
to this segment. Several AIME problems have shown me triangles with nice side lengths. For this one, the cute triangle was 
. You can get it from taking the colloquial 
, and then removing a 
.
and find 
, 
, 
, and 
in terms of 
. Find using 
. Then use Heron's on 
and multiply by 
to get ![\[168\cdot \frac{39}{14}=\boxed{468}.\]](http://latex.artofproblemsolving.com/e/2/b/e2b9aba05c08b1bdc4c4069860dedfcaccac222a.png)
as well using herons formula to find the height.
.
such that the intersection is between C and D (we will call that . So, by herons, we have
.
and 
.
,
.
