IMO 1979 Problem B2 Find all real numbers a for which there exist non-negative real numbers x 1 , x 2 , x 3 , x 4 , x 5 satisfying: x 1 + 2x 2 + 3x 3 + 4x 4 + 5x 5 = a, x 1 + 2 3 x 2 + 3 3 x 3 + 4 3 x 4 + 5 3 x 5 = a 2 , x 1 + 2 5 x 2 + 3 5 x 3 + 4 5 x 4 + 5 5 x 5 = a 3 .
Take a 2 x 1st equ - 2a x 2nd equ + 3rd equ. The rhs is 0. On the lhs the coefficient of x n is a 2 n - 2an 3 + n 5 = n(a - n 2 ) 2 . So the lhs is a sum of non-negative terms. Hence each term must be zero separately, so for each n either x n = 0 or a = n 2 . So there are just 5 solutions, corresponding to a = 1, 4, 9, 16, 25. We can check that each of these gives a solution. [For a = n 2 , x n = n and the other x i are zero.] s are also available in Murray S Klamkin, International Mathematical Olympiads 1978-1985, MAA 1986, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48- X. 21st IMO 1979 © John Scholes jscholes@kalva.demon.co.uk 12 Oct 1998