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P^a+Q^b=R^c and 1/a+1/b+1/c>1, polynomials

Source: Mathcenter Contest / Oly - Thai Forum 2010 R1 p1 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 15, 2022
polynomialalgebra

Problem Statement

Let a,b,c∈Na,b,c\in\mathbb{N} prove that if there is a polynomial P,Q,R∈C[x]P,Q,R\in\mathbb{C}[x], which have no common factors and satisfy Pa+Qb=RcP^a+Q^b=R^c and \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}>1.
(tatari/nightmare)
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