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Polynomials P(x), Q(x) with a_n-b_n as prime [INMO 2011]

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February 6, 2011
algebrapolynomialcalculusintegrationRational Root Theoremnumber theory

Problem Statement

Let P(x)=anxn+an−1xn−1+⋯+a0P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0 and Q(x)=bnxn+bn−1xn−1+⋯+b0Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0 be two polynomials with integral coefficients such that an−bna_n-b_n is a prime and anb0−a0bn≠0,a_nb_0-a_0b_n\neq 0, and an−1=bn−1.a_{n-1}=b_{n-1}. Suppose that there exists a rational number rr such that P(r)=Q(r)=0.P(r)=Q(r)=0. Prove that r∈Z.r\in\mathbb Z.
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