1st Iberoamerican 1985 Problem A3 Find the roots r 1 , r 2 , r 3 , r 4 of the equation 4x 4 - ax 3 + bx 2 - cx + 5 = 0, given that they are positive reals satisfying r 1 /2 + r 2 /4 + r 3 /5 + r 4 /8 = 1.
We have r 1 r 2 r 3 r 4 = 5/4 and hence (r 1 /2) (r 2 /4) (r 3 /5) (r 4 /8) = 1/4 4 . But AM/GM gives that (r 1 /2) (r 2 /4) (r 3 /5) (r 4 /8) ≤ ( (r 1 /2 + r 2 /4 + r 3 /5 + r 4 /8)/4 ) 4 = 1/4 4 with equality iff r 1 /2 = r 2 /4 = r 3 /5 = r 4 /8. Hence we must have r 1 = 1/2, r 2 = 1, r 3 = 5/4, r 4 = 2. 1st Ibero 1985 © John Scholes jscholes@kalva.demon.co.uk 1 July 2002