2018 homc The center of a circle and nine randomly selected points on this circle are colored in red. Every pair of those points is connected by a line segment, and every point of intersection of two line segments inside the circle is colored in red. What is the largest possible number of red points?

Max possible red points  = 220

Step-by-step explanation:

- Remark that a convex quadrilateral has exactly one intersection which is the intersection of its two diagonals.

- Consider 9 points on the circle, which give at most ( combination ):

                            intersections.

- Now consider the center and the three points on the circle, there are at most ( combinations ):

                            intersections.

- So the total number of red points would be:

                           Given = 10

                           2 Diagonals = 126

                           Center + 3 pts = 84

                         ======================

            Max possible red points  = 220

                         ======================