19th Swedish 1979 Problem 1 Solve the equations: x 1 + 2 x 2 + 3 x 3 + ... + (n-1) x n-1 + n x n = n 2 x 1 + 3 x 2 + 4 x 3 + ... + n x n-1 + x n = n-1 3 x 1 + 4 x 2 + 5 x 3 + ... + x n-1 + 2 x n = n-2 ... (n-1) x 1 + n x 2 + x 3 + ... + (n-3) x n-1 + (n-2) x n = 2 n x 1 + x 2 + 2 x 3 + ... + (n-2) x n-1 + (n-1) x n = 1.
x 1 = 2/n - 1, x 2 = x 3 = ... = x n = 2/n Solution Adding the equations we get x 1 + x 2 + ... + x n = 1. Subtracting each equation from the following equation we get: x 1 + x 2 + ... + x n - nx k = -1 for k = 2, ... , n. Hence x k = 2/n for k = 2, 3, ... , n and hence x 1 = 2/n - 1. Thanks to Thomas Linhart 19th Swedish 1979 © John Scholes jscholes@kalva.demon.co.uk 5 March 2004 Last corrected/updated 5 Mar 04