28th CanMO 1996 Problem 1 The roots of x 3 - x - 1 = 0 are r, s, t. Find (1 + r)/(1 - r) + (1 + s)/(1 - s) + (1 + t)/(1 - t).
Put y = (1 + x)/(1 - x). Then x = (y - 1)/(y + 1), so y satisfies (y - 1) 3 - (y - 1)(y + 1) 2 - (y + 1) 3 = 0 or y 3 + 7y 2 - y + 1 = 0. So the sum of the roots is -7. 28th CanMO 1996 © John Scholes jscholes@kalva.demon.co.uk 15 June 2002