Question 109
Let $r$ be some positive real number. It is known that for some positive integer $n$ the following condition holds: all positive real numbers $a_1, \dots, a_n$ satisfying the equality $a_1 + \dots + a_n = r\left(\frac{1}{a_1} + \dots + \frac{1}{a_n}\right)$, $a_i \ne \sqrt{r}$, $i = 1, \dots, n$, also satisfy the equality $\frac{1}{\sqrt{r} - a_1} + \dots + \frac{1}{\sqrt{r} - a_n} = \frac{1}{\sqrt{r}}$. Find $n$.
(D. Bazylev)
(D. Bazylev)