12th USAMO 1983 Problem 1 If six points are chosen sequentially at random on the circumference of a circle, what is the probability that the triangle formed by the first three is disjoint from that formed by the second three. Solution
: 3/10. Only the order is important. We are interested in permutations of 123456 where the 123 are together (allowing wrapping). wlog the 1 is in first position. So the triangles are disjoint in the cases 123xxx, 132xxx, 1xxx23, 1xxx32, 12xxx3, 13xxx2. So the probability is 6·6/5! = 3/10. 12th USAMO 1983 © John Scholes jscholes@kalva.demon.co.uk 24 Aug 2002