Try this beautiful problem from HANOI, 2018 based on **Circles and Points.**

## Circles – HANOI 2018

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.

- is 224
- is 220
- is 228
- cannot be determined from the given information

**Key Concepts**

Geometry

Circles

Combination

## Check the Answer

But try the problem first…

Answer: is 220.

HANOI, 2018

Geometry Revisited by Coxeter

## Try with Hints

First hint

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.

Second Hint

Considering the center and three points on the circle, there are at most \(\frac{9!}{3!6!}\)=84 intersections.

Final Step

So there are at most 126+84+10=220 red points.

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s