## Task 3 - 090933

For a certain task, A needs exactly m times as long as B and C together; B needs exactly n times as long as C and A together, and C exactly p times as long as A and B together. Calculate $p$ in terms of $m$ and $n$!
We consider the performances $a, b$, and $c$ of $A, B$, and $C$. Then, according to the problem statement,

$$
a=\frac{1}{m}(b+c) ; \quad b=\frac{1}{n}(a+c) ; \quad c=\frac{1}{p}(a+b)
$$

The first equation is equivalent to $c=a m-b$, the second to $c=b n-a$. In particular, therefore, $a m-b=b n-a$ or $(m+1) a=(n+1) b$, thus

$$
b=\frac{m+1}{n+1} \cdot a \quad \text { and } \quad c=a m-b=\frac{m n-1}{n+1} \cdot a
$$

If $m n-1=0$, i.e., $c=0$, then $C$ does not perform any work and thus $p$ is not defined. Otherwise, we can divide by $c \neq 0$ in the following.

Using the third equation, we finally obtain by substitution

$$
p=\frac{a+b}{c}=\frac{\frac{n+1+m+1}{n+1} \cdot a}{\frac{m n-1}{n+1} \cdot a}=\frac{m+n+2}{m n-1}
$$