OMCC 2001 Problem 6

In a circumference of a circle, $10000$ points are marked, and they are numbered from $1$ to $10000$ in a clockwise manner. $5000$ segments are drawn in such a way so that the following conditions are met:

1. Each segment joins two marked points.
2. Each marked point belongs to one and only one segment.
3. Each segment intersects exactly one of the remaining segments.
4. A number is assigned to each segment that is the product of the number assigned to each end point of the segment.

Let $S$ be the sum of the products assigned to all the segments.

Show that $S$ is a multiple of $4$ .