40th VMO 2002 Problem Solve the following equation: √(4 - 3√(10 - 3x)) = x - 2. Solution
: x = 3. Squaring twice we get x 4 - 8x 3 + 16x 2 + 9x - 90 = 0. Factorising, we get (x + 2)(x - 3)(x 2 - 7x + 15), so the only real roots are x = -2 and 3. Checking, we find that 3 is indeed a solution of the original equation, but x = -2 is not because we get √(-8) on the lhs. Thanks to Suat Namli for a similar solution. 40th VMO 2002 © John Scholes jscholes@kalva.demon.co.uk 21 Nov 2002 Last updated/corrected 21 Nov 02