7th CanMO 1975 Problem 4 Find all positive real x such that x - [x], [x], x form a geometric progression.
Let [x] = n. We have x - n  2, but x/n < 2. We must have n ≥ 1, otherwise n < x - n. Hence n = 1. Let x = 1 + y. Then 1/y = 1 + y, so y = (√5 - 1)/2 and x = (√5 + 1)/2. 7th CanMO 1975 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002