$3 \cdot 44$ To complete the same work, the time required for $A$ to work alone is $m$ times the time required for $B$ and $C$ to work together; the time required for $B$ to work alone is $n$ times the time required for $A$ and $C$ to work together; and the time required for $C$ to work alone is $k$ times the time required for $A$ and $B$ to work together; then $k$ expressed in terms of $m, n$ is
(A) $\frac{2 m n}{m+n}$.
(B) $\frac{1}{m+n-m n}$.
(C) $\frac{1-m n}{m+n+2 m n}$.
(D) $\frac{m+n+2}{m n-1}$.
(2nd "Jinyun Cup" Junior High School Mathematics Invitational Competition, 1985)
[Solution] Let the daily workloads of $A, B, C$ be $x, y, z$, respectively. According to the problem, we have
$$
\begin{array}{l}
\left\{\begin{array} { l } 
{ \frac { 1 } { x } = \frac { m } { y + z } , } \\
{ \frac { 1 } { y } = \frac { n } { x + z } . }
\end{array} \text { i.e., } \quad \left\{\begin{array}{l}
m x-y=z, \\
n y-x=z .
\end{array}\right.\right. \\
\left\{\begin{array}{l}
x=\frac{n+1}{m n-1} z, \\
y=\frac{m+1}{m n-1} z .
\end{array}\right.
\end{array}
$$

From $\frac{1}{z}=\frac{k}{x+y}$, we get $k=\frac{x+y}{z}=\frac{m+n+2}{m n-1}$. Therefore, the answer is $(D)$.