How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

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




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.

Subscribe to Cheenta at Youtube

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.