1983 USAMO Problems/Problem 1

Problem

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?

Hint

First choose six points on the circumference on a circle. What is the probability that selecting three random points will yield a triangle that is disjoint from the triangle formed by the other three points?

Solution

The probability in the hint equals , and that is the answer! The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.