RO-TST 2005 Problem 15

Let $n\geq 2$ be an integer. Find the smallest real value $\rho (n)$ such that for any $x_i>0$ , $i=1,2,\ldots,n$ with $x_1 x_2 \cdots x_n = 1$ , the inequality $$\sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r$$ is true for all $r\geq \rho (n)$ .