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Polynomial inequality

Source: Centroamerican 2020, problem 5

October 28, 2020
algebrapolynomialInequalityinequalitiesn-variable inequality

Problem Statement

Let P(x)P(x) be a polynomial with real non-negative coefficients. Let kk be a positive integer and x1,x2,…,xkx_1, x_2, \dots, x_k positive real numbers such that x1x2⋯xk=1x_1x_2\cdots x_k=1. Prove that
P(x1)+P(x2)+⋯+P(xk)≥kP(1).P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).
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