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# Circles and points | HANOI 2018

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

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.

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