IMO 1966 Problem B2 Solve the equations: |a i - a 1 | x 1 + |a i - a 2 | x 2 + |a i - a 3 | x 3 + |a i - a 4 | x 4 = 1, i = 1, 2, 3, 4, where a i are distinct reals.
x 1 = 1/(a 1 - a 4 ), x 2 = x 3 = 0, x 4 = 1/(a 1 - a 4 ). Solution Take a 1 > a 2 > a 3 > a 4 . Subtracting the equation for i=2 from that for i=1 and dividing by (a 1 - a 2 ) we get: - x 1 + x 2 + x 3 + x 4 = 0. Subtracting the equation for i=4 from that for i=3 and dividing by (a 3 - a 4 ) we get: - x 1 - x 2 - x 3 + x 4 = 0. Hence x 1 = x 4 . Subtracting the equation for i=3 from that for i=2 and dividing by (a 2 - a 3 ) we get: - x 1 - x 2 + x 3 + x 4 = 0. Hence x 2 = x 3 = 0, and x 1 = x 4 = 1/(a 1 - a 4 ). Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48- X. 8th IMO 1966 © John Scholes jscholes@kalva.demon.co.uk 29 Sep 1998 Last corrected/updated 26 Sep 2003