1st Junior Balkan 1997 Problem 2 Given reals x, y with (x 2 + y 2 )/(x 2 - y 2 ) + (x 2 - y 2 )/(x 2 + y 2 ) = k, find (x 8 + y 8 )/(x 8 - y 8 ) + (x 8 - y 8 )/(x 8 + y 8 ) in terms of k.
k/2 = (x 4 +y 4 )/(x 4 -y 4 ), so k/2 + 2/k = 2(x 8 +y 8 )/(x 8 -y 8 ). Hence desired expresson = (k 2 +4)/4k + 4k/(k 2 +4) = (k 4 +24k 2 +16)/(4k 3 +14k). Thanks to Cristian Ilac 1st Junior Balkan 1997 © John Scholes jscholes@kalva.demon.co.uk 3 February 2004 Last corrected/updated 3 Feb 04