19th ASU 1985 Problem 20 x is a real number. Define x 0 = 1 + √(1 + x), x 1 = 2 + x/x 0 , x 2 = 2 + x/x 1 , ... , x 1985 = 2 + x/x 1984 . Find all solutions to x 1985 = x.
3 Solution If x = 0, then x 1985 = 2 ≠ x. Otherwise we find x 1 = 2 + x/(1+√(1+x)) = 2 + (√(1+x) - 1) = 1 + √(1+x). Hence x 1985 = 1 + √(1+x). So x - 1 = √(1+x). Squaring, x = 0 or 3. We have already ruled out x = 0. It is easy to check that x = 3 is a solution. Thanks to Suat Namli 19th ASU 1985 © John Scholes jscholes@kalva.demon.co.uk 16 March 2004 Last corrected/updated 16 Mar 04