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Find the greatest lambda so that the second inequality holds

Source: China TST 2011 - Quiz 2 - D2 - P3

May 20, 2011
inequalitiesinequalities unsolved

Problem Statement

Let nn be a positive integer. Find the largest real number λ\lambda such that for all positive real numbers x1,x2,⋯ ,x2nx_1,x_2,\cdots,x_{2n} satisfying the inequality 12n∑i=12n(xi+2)n≥∏i=12nxi,\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i, the following inequality also holds 12n∑i=12n(xi+1)n≥λ∏i=12nxi.\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.
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