Triangles Containing the Origin II

Define:$x_n = (1248^n \bmod 32323) – 16161$$y_n = (8421^n \bmod 30103) – 15051$
$P_n = \{(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\}$

For example, $P_8 = \{(-14913, -6630),$$(-10161, 5625),$$(5226, 11896),$$(8340, -10778),$$(15852, -5203),$$(-15165, 11295),$$(-1427, -14495),$$(12407, 1060)\}$.
Let $C(n)$ be the number of triangles whose vertices are in $P_n$ which contain the origin in the interior.

$C(8) = 20$
$C(600) = 8950634$
$C(40\,000) = 2666610948988$

Find $C(2\,000\,000)$.

Given the complexity of the model and the size of the requested output, this problem falls into the realm of competitive programming and it is certainly a non-trivial task.

The step-by-step solution to such kind of problems typically involves programming languages (such as Python, C++, Java, etc.). Some mathematical expertise and algorithmic knowledge are also essential to efficiently process large quantities of data and build an adequate model to solve this problem.

Below is a generalized approach, but please note that the exact programming code is beyond the capability of this language model.

General Approach:

1. Compute the points `P_n` for n up to 2,000,000 by iterating n from 1 to 2,000,000 and calculating `x_n` and `y_n` by exponentiation modulo 32323 for `x` and 30103 for `y`. For this type of computation, it is recommended to use modular exponentiation to handle the large power.

2. Once the full set of points is calculated, iterate over each triplet of points (forming triangles). For each triangle, check if the origin lies within it.

3. To check if the origin (O) lies within a triangle, you can use Barycentric coordinates or simply compute the area of the triangle ABC and compare it to the sum of the areas of the triangles OAB, OBC and OCA. If the areas are equal, the origin lies within the triangle.

4. Count the number of triangles meeting the criteria (i.e., the origin lies within) and that should give you `C(2,000,000)`.

Note: Given the high computational requirements, especially for checking every triangle, optimizing this process requires usage of computational geometry algorithms, efficient data structures (for example a sweep line approach), and perhaps parallel programming techniques.

More Answers:
Long Products
Lattice Quadrilaterals
Diophantine Reciprocals III

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