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Turkey NMO 2000 Problem 2

Source: Turkey NMO 2000 Problem 2

September 29, 2011
algebrapolynomialnumber theoryrelatively primealgebra proposed

Problem Statement

Let define Pn(x)=xn−1+xn−2+xn−3+⋯+x+1P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1 for every positive integer nn. Prove that for every positive integer aa one can find a positive integer nn and polynomials R(x)R(x) and Q(x)Q(x) with integer coefficients such that Pn(x)=[1+ax+x2R(x)]Q(x).P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).
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