17th VMO 1979 Problem A2 Find all real numbers a, b, c such that x 3 + ax 2 + bx + c has three real roots α,β,γ (not necessarily all distinct) and the equation x 3 + a 3 x 2 + b 3 x + c 3 has roots α 3 , β 3 , γ 3 .
(a,b,c) = (h,k,hk), where k ≤ 0 Solution We have -a = α + β + γ, b = αβ + βγ + γα, -c = αβγ. We need -a 3 = α 3 + β 3 + γ 3 . Since -a 3 = (α + β + γ) 3 = α 3 + β 3 + γ 3 + 3(αβ 2 + ... ) + 6αβγ = -a 3 - 3ab + 3c, we require c = ab. Now if c = ab, the cubic factorises as (x+a)(x 2 +b), so we get three real roots iff b ≤ 0. But if b ≤ 0 and c = ab, then the roots are -a, ±√(-b). So the equation with roots α 3 , β 3 , γ 3 is (x + a 3 )(x 2 + b 3 ) = x 3 + a 3 x 2 + b 3 x + a 3 b 3 . Thus c = ab, b ≤ 0 is both a necessary and a sufficient condition. 17th VMO 1979 © John Scholes jscholes@kalva.demon.co.uk 6 March 2004 Last corrected/updated 6 Mar 04