MathDB

5

Part of 2020 Centroamerican and Caribbean Math Olympiad

Problems(1)

Polynomial inequality

Source: Centroamerican 2020, problem 5

10/28/2020
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).
algebrapolynomialInequalityinequalitiesn-variable inequality