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P(1/x) * P(x) >= 1 for x = 1 ==> also for every x > 0

Source: Arthur Engel, intended German TST problem 2004; Baltic Way 2004, problem 2

May 1, 2004
algebrapolynomialinequalitiessearchalgebra solved

Problem Statement

Let P(x) P(x) be a polynomial with a non-negative coefficients. Prove that if the inequality P(1x)P(x)≥1 P\left(\frac {1}{x}\right)P(x)\geq 1 holds for x \equal{} 1, then this inequality holds for each positive x x.
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