Try this beautiful problem from HANOI, 2018 based on Circles and Points.
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. Find the largest possible number of red points.
Geometry
Circles
Combination
Answer: is 220.
HANOI, 2018
Geometry Revisited by Coxeter
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 \(\frac{9!}{4!5!}\)=126 intersections.
Considering the center and three points on the circle, there are at most \(\frac{9!}{3!6!}\)=84 intersections.
So there are at most 126+84+10=220 red points.
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