8th Swedish 1968 Problem 1 Find the maximum and minimum values of x 2 + 2y 2 + 3z 2 for real x, y, z satisfying x 2 + y 2 + z 2 = 1.
3, 1 Solution x 2 + 2y 2 + 3z 2 = (x 2 + y 2 + z 2 ) + y 2 + 2z 2 = 1 + y 2 + 2z 2 ≥ 1. Equality if x = 1, y = z = 0. x 2 + 2y 2 + 3z 2 ≤ 3(x 2 + y 2 + z 2 ) = 3. Equality if x = y = 0, z = 1. 8th Swedish 1968 © John Scholes jscholes@kalva.demon.co.uk 30 December 2003 Last corrected/updated 30 Dec 03