IMO 1992 LL VIE82

Origin: VIE

Problem

Let f(x) = xm + a1xm−1 + \cdot \cdot \cdot + am−1x + am and g(x) = xn + b1xn−1 + \cdot \cdot \cdot + bn−1 + bn be two polynomials with real coefficients such that for each real number x, f(x) is the square of an integer if and only if so is g(x). Prove that if n + m > 0, then there exists a polynomial h(x) with real coefficients such that f(x) \cdot g(x) = (h(x))2.