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Triangles from polynoms

Source: All Russian Olympiad 2017,Day2,grade 11,P5

May 1, 2017
algebrapolynomialDecreasing

Problem Statement

P(x)P(x) is polynomial with degree n≥2n\geq 2 and nonnegative coefficients. a,b,ca,b,c - sides for some triangle. Prove, that P(a)n,P(b)n,P(c)n\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)} are sides for some triangle too.
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