CN-TST 2011 Problem 12

Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality $$\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,$$ the following inequality also holds $$\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.$$