*A writeup of Dec number 1 from the USACO platinum problem set*

### Lots of Triangles

#### Problem Statement

Farmer John is thinking of selling some of his land to earn a bit of extra income. His property contains n trees (3≤N≤300), each described by a point in the 2D plane, no three of which are collinear. FJ is thinking about selling triangular lots of land defined by having trees at their vertices; A triangular lot has value v if it contains exactly v trees in its interior (the trees on the corners do not count, and note that there are no trees on the boundaries since no three trees are collinear). For every v=0…N−3, please help FJ determine how many of his L potential lots have value v.

#### Observations

For all pairs of trees, we store the number of trees under this line. Because no three trees are collinear, none will overlap with this line. This takes O(N^3) time.

Observe that by doing this, we can find the number of trees within an arbitrary triangle. We do this by adding or subtracting the number of trees under each edge.

#### Solution

More specifically, we add the value if the edge is above last remaining point in the triangle, and we subtract it if it is below.

Doing this for O(N^3) triangles finishes.

My solution can be found here

*Written by Robert Chen*

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