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March 18, 2020

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

Check the Answer


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.

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