Question 1. The sum of $\mathrm{n}$ positive numbers is 1983. When is their product maximized, and what is the maximum value?
Since the geometric mean of $n$ positive numbers is not greater than their arithmetic mean, when the product of $n$ positive numbers $a_{1}, a_{2}, \cdots, a_{n}$ is 1983,
$$
\begin{array}{l}
a_{1} a_{2} \cdots a_{n} \leqslant\left(\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)^{n} \\
=\left(\frac{1983}{n}\right)^{n}
\end{array}
$$

The equality holds if and only if $a_{1}=a_{2}=\cdots=a_{n}=\frac{1983}{n}$.

Therefore, the maximum value sought is $\left(\frac{1983}{n}\right)$, which is achieved when all $n$ numbers are equal to $\frac{1983}{n}$.