6th Irish 1993 Problem A1 The real numbers x, y satisfy x 3 - 3x 2 + 5x - 17 = 0, y 3 - 3y 2 + 5y + 11 = 0. Find x + y.
2 Solution x 3 -3x 2 +5x = (x-1) 3 +2(x-1)+3, which is a strictly monotonic increasing function. Hence x 3 -3x 2 +5x-17 = 0 has just one real root. Similarly so does y 3 -3y 2 +5y+11 = 0. If we substitute x = 2-y in x 3 -3x 2 +5x-17 we get -(y 3 -3y 2 +5y+11). Hence the two roots satisfy x+y = 2. 6th Irish 1993 © John Scholes jscholes@kalva.demon.co.uk 6 February 2004 Last updated/corrected 6 Feb 04