Problem

Source: China Nanchang

Tags: inequalities



Let $n$ be positive integer,$x_1,x_2,\cdots,x_n$ be positive real numbers such that $x_1x_2\cdots x_n=1 $ . Prove that$$\sum\limits_{i = 1}^{n}x_i\sqrt{x^2_1+x^2_2+\cdots x^2_i}\ge\frac{n+1}{2}\sqrt{n}$$