IMO 1978 LL NET31

Origin: NET

Problem

Let the polynomials P(x) = xn + an−1xn−1 + \cdot \cdot \cdot + a1x + a0, Q(x) = xm + bm−1xm−1 + \cdot \cdot \cdot + b1x + b0, be given satisfying the identity P(x)2 = (x2 −1)Q(x)2 + 1. Prove the identity P ′(x) = nQ(x).