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Polynomials - IMO LongList 1992 VIE1 (Last Problem of ILL)

Source:

September 2, 2010
algebrapolynomialIMO ShortlistProduct

Problem Statement

Let f(x)=xm+a1xm−1+⋯+am−1x+amf(x) = x^m + a_1x^{m-1} + \cdots+ a_{m-1}x + a_m and g(x)=xn+b1xn−1+⋯+bn−1x+bng(x) = x^n + b_1x^{n-1} + \cdots + b_{n-1}x + b_n be two polynomials with real coefficients such that for each real number x,f(x)x, f(x) is the square of an integer if and only if so is g(x)g(x). Prove that if n+m>0n +m > 0, then there exists a polynomial h(x)h(x) with real coefficients such that f(x)⋅g(x)=(h(x))2.f(x) \cdot g(x) = (h(x))^2.
[hide="Remark."]Remark. The original problem stated g(x)=xn+b1xn−1+⋯+bn−1+bng(x) = x^n + b_1x^{n-1} + \cdots + {\color{red}{ b_{n-1}}} + b_n, but I think the right form of the problem is what I wrote.
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