A regular 201-sided polygon is inscribed inside a circle of center C. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point C lying inside the triangle?

There are 201 ways of choosing the first point.

Make the second point be counterclockwise from the first point and the third clockwise. There are 100 points to choose from for the second point, but each has a different number of valid third points that go along with it.

The farthest possible second point allows 100 choices for the third. The next possible second point allows only 99, etc. all the way down to the point adjacent to the first point, which allows only one choice of third point. 

So for a given chosen first point, there are 100 + 99 + ... + 2 + 1 = 5050 possibilities for the other two points.

Since there are 201 choices for the first point, but any one of three vertices can be considered the first point, the required answer is 201 * 5050 / 3 = 338,350.

Posted by Charlie on 2014-02-27 12:39:52