If $n>2$ is an integer and $x_1, \ldots ,x_n$ are positive reals such that \[ \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n} = n \]then the following inequality takes place \[ \frac{x_2^2+\cdots+x_n^2}{n-1}\cdot \frac {x_1^2+x_3^2+\cdots +x_n^2} {n-1} \cdots \frac{x_1^2+\cdots+x_{n-1}^2}{n-1}\geq \left(\frac{x_1^2+...+x_n^2}{n}\right)^{n-1}. \]