Question

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?

Answer 1

Max possible red points = 220

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 ):

`(9!)/(4!5!) = 126` intersections.

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

`(9!)/(3!6!) = 84` intersections.

- So the total number of red points would be:

Given = 10

2 Diagonals = 126

Center + 3 pts = 84

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Max possible red points = 220

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