Let f(x)=xm+a1xm−1+⋯+am−1x+am and g(x)=xn+b1xn−1+⋯+bn−1x+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)⋅g(x)=(h(x))2.[hide="Remark."]Remark. The original problem stated g(x)=xn+b1xn−1+⋯+bn−1+bn, but I think the right form of the problem is what I wrote. algebrapolynomialIMO ShortlistProduct