7th Junior Balkan 2003 Problem 2 A 1 , A 2 , ... , A n are points in the plane, so that if we take the points in any order B 1 , B 2 , ... , B n , then the broken line B 1 B 2 ...B n does not intersect itself. What is the largest possible value of n?
4 Solution We cannot have three points collinear, because if B lies on the segment AC, then the broken line ...ACB... intersects itself. If we take any four points, then their convex hull must be a triangle. This works for 4 points. Suppose we have 5 points A, B, C, D, E. Then we can take D to be inside the triangle ABC. But there is now nowhere to put E. If it is in 1, then ABDE is self-intersecting. Similarly, in 2, ACDE is self-intersecting, in 3 BCDE is self-intersecting, in 4 ADCE is self intersecting and so on. Thanks to Suat Namli 7th Junior Balkan 2003 © John Scholes jscholes@kalva.demon.co.uk 7 February 2004 Last corrected/updated 7 Feb 04