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Israel 2013 Q6 - Fractional inequality

Source: Israel National Olympiad 2013 Q6

August 8, 2019
algebraInequalityFractionsinequalities

Problem Statement

Let x1,...,xnx_1,...,x_n be positive real numbers, satisfying x1+⋯+xn=nx_1+\dots+x_n=n. Prove that
x1x2+x2x3+⋯+xn−1xn+xnx1≤4x1⋅x2⋅⋯⋅xn+n−4\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4.
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