IMO 1984 LL LUX31

Origin: LUX

Problem

Let f1(x) = x3 +a1x2 +b1x+c1 = 0 be an equation with three positive roots \alpha > \beta > \gamma > 0. From the equation f1(x) = 0 one constructs the equation f2(x) = x3 + a2x2 + b2x + c2 = x(x + b1)2 −(a1x + c1)2 = 0. Continuing this process, we get equations f3, . . . , fn. Prove that lim n\to\infty 2n−1\sqrt−an = \alpha.