30th IMO 1989 shortlist Problem 20 b > 0 and a are fixed real numbers and n is a fixed positive integer. The real numbers x 0 , x 1 , ... , x n satisfy x 0 + x 1 + ... + x n = a and x 0 2 + ... + x n 2 = b. Find the range of x 0 .
By Cauchy we have (∑ x i ) 2 ≤ n ∑ x i 2 (for n terms) (*). Hence if x 0 = x, then (a - x) 2 ≤ n(b - x 2 ) or (n+1) x 2 - 2ax + a 2 - nb ≤ 0. If a 2  0. 30th IMO shortlist 1989 © John Scholes jscholes@kalva.demon.co.uk 28 Dec 2002 Last corrected/updated 28 Dec 02