IMO 1959 Problem A2 For what real values of x is √(x + √(2x-1)) + √(x - √(2x-1)) = A, given (a) A = √2, (b) A = 1, (c) A = 2, where only non-negative real numbers are allowed in square roots and the root always denotes the non- negative root?
(a) any x in the interval [1/2,1]; (b) no solutions; (c) x=3/2. Solution Note that we require x ≥ 1/2 to avoid a negative sign under the inner square roots. Since (x-1) 2 ≥ 0, we have x ≥ √(2x-1), so there is no difficulty with √(x - √(2x-1)), provided that x ≥ 1/2. Squaring gives 2x + 2√(x 2 -2x+1) = A 2 . Note that the square root is |x-1|, not simply (x-1). So we get finally 2x + 2|x-1| = A 2 . It is now easy to see that we get the solutions above. Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48- X. 1st IMO 1959 © John Scholes jscholes@kalva.demon.co.uk 17 Sep 1998 Last corrected/updated 26 Jan 2004