1st CaMO 1969 Problem 2 If x is a real number not less than 1, which is larger: √(x+1) - √x or √x - √(x-1)? Solution
: √x - √(x-1) is larger. 4x 2 - 4x + 1 > 4x 2 - 4x. Taking the positive square root of each side gives 2x - 1> 2√(x 2 -x). Hence x 2 + x > x 2 - x + 1 + 2√(x 2 -x). Taking the positive square root, we get √(x 2 +x) > √(x 2 -x) + 1. Hence x + (x - 1) - 2√(x 2 -x) > (x + 1) + x - 2√(x 2 +x). Taking the positive square root, √x - √(x-1) > √(x+1) - √x. 1st CaMO 1969 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002