Example 1. Let positive numbers $a_{1}, a_{2}, \cdots, a_{\mathrm{n}}$ satisfy $a_{1}+a_{2}+\cdots+a_{\mathrm{n}}=1$. Find
$$
\begin{array}{l}
\frac{a_{1}}{1+a_{2}+\cdots+a_{\mathrm{n}}}+\frac{a_{2}}{1+a_{1}+a_{3}+\cdots+a_{n}} \\
+\cdots+\frac{a_{1}}{1+a_{1}+\cdots+a_{\mathrm{n}-1}}
\end{array}
$$

the minimum value. (82 West Germany Competition Problem)