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Inequality on n numbers - sum is geq 1/n(alpha+beta)

Source: Moldova TST 2002 - E2 - P1

January 8, 2012
inequalitiesinequalities unsolved

Problem Statement

Positive numbers \alpha ,\beta , x_1, x_2,\ldots, x_n (n≥1n \geq 1) satisfy x1+x2+⋯+xn=1x_1+x_2+\cdots+x_n = 1. Prove that ∑i=1nxi3αxi+βxi+1≥1n(α+β).\sum_{i=1}^{n} \frac{x_i^3}{\alpha x_i+\beta x_{i+1}} \geq \frac{1}{n(\alpha+\beta)}. Note. xn+1=x1x_{n+1}=x_1.
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