### Problem Statement

A polygon is a figure on a plane bounded by a closed polyline with no self-intersections. A diagonal is a line segment that connects two non-adjacent vertices and lies completely inside the polygon. Every diagonal divides the polygon into two other polygons.

A polygon is called convex if every line segment that connects two non-adjacent vertices is a diagonal. A polygon is called almost convex if it is convex or if there exists a diagonal that divides it into two convex polygons.

You are given a set of points. The coordinates of the i-th point are (x[i], y[i]). Return the number of distinct almost convex polygons whose vertices all belong to the given set.

### Definition

 Class: AlmostConvexPolygons Method: count Parameters: int[], int[] Returns: int Method signature: int count(int[] x, int[] y) (be sure your method is public)

### Notes

-The answer will always fit in 32-bit signed integer.

### Constraints

-x will contain between 3 and 15 elements, inclusive.
-y will contain the same number of elements as x.
-Each element of x and y will be between -100 and 100, inclusive.
-No three points from the given set will lie on the same straight line.

### Examples

0)

 `{-2,2,-2,2}` `{-2,-2,2,2}`
`Returns: 5`
 We can construct 4 triangles and 1 square. All these polygons are convex and therefore are almost convex.
1)

 `{-2,2,-2,2,3}` `{-2,-2,2,2,0}`
`Returns: 16`
 Point (3, 0) is added to example 0. Now we can construct 10 triangles, 5 quadrangles and 1 pentagon. All of them are still convex.
2)

 `{-2,2,-2,2,0}` `{-2,-2,2,2,1}`
`Returns: 22`
 Point (0, 1) is added to example 0. We can construct 10 triangles, 9 quadrangles and 4 pentagons. All of them are almost convex except one pentagon (-2,-2) - (0,1) - (2,-2) - (2,2) - (-2,2) - (-2,-2).
3)

 `{0,5,0,2,1}` `{0,0,5,1,2}`
`Returns: 26`
 Here there are 10 triangles, 13 almost convex quadrangles and 3 almost convex pentagons: (0,0) - (2,1) - (5,0) - (0,5) - (1,2) - (0,0); (0,0) - (5,0) - (0,5) - (1,2) - (2,1) - (0,0); (0,0) - (1,2) - (2,1) - (5,0) - (0,5) - (0,0).

#### Problem url:

http://www.topcoder.com/stat?c=problem_statement&pm=8768

#### Problem stats url:

http://www.topcoder.com/tc?module=ProblemDetail&rd=12018&pm=8768

ivan_metelsky

#### Testers:

PabloGilberto , SnapDragon , Olexiy , Mike Mirzayanov

#### Problem categories:

Brute Force, Geometry