11. (GDR 1) Given real numbers $x_{1}, x_{2}, \ldots, x_{n}(n \geq 2)$, with $x_{i} \geq 1 / n$ $(i=1,2, \ldots, n)$ and with $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$, find whether the product $P=x_{1} x_{2} x_{3} \cdots x_{n}$ has a greatest and/or least value and if so, give these values.
11. Let us define $y_{i}=x_{i}^{2}$. We thus have $y_{1}+y_{2}+\cdots+y_{n}=1, y_{i} \geq 1 / n^{2}$, and $P=\sqrt{y_{1} y_{2} \ldots y_{n}}$. The upper bound is obtained immediately from the AM-GM inequality: $P \leq 1 / n^{n / 2}$, where equality holds when $x_{i}=\sqrt{y_{i}}=1 / \sqrt{n}$. For the lower bound, let us assume w.l.o.g. that $y_{1} \geq y_{2} \geq \cdots \geq y_{n}$. We note that if $a \geq b \geq 1 / n^{2}$ and $s=a+b>2 / n^{2}$ is fixed, then $a b=\left(s^{2}-(a-b)^{2}\right) / 4$ is minimized when $|a-b|$ is maximized, i.e., when $b=1 / n^{2}$. Hence $y_{1} y_{2} \cdots y_{n}$ is minimal when $y_{2}=y_{3}=\cdots=y_{n}=1 / n^{2}$. Then $y_{1}=\left(n^{2}-n+1\right) / n^{2}$ and therefore $P_{\min }=\sqrt{n^{2}-n+1} / n^{n}$.