Problem 8.7. For three real numbers $p, q$, and $r$, it is known that

$$
p+q+r=5, \quad \frac{1}{p+q}+\frac{1}{q+r}+\frac{1}{p+r}=9
$$

What is the value of the expression

$$
\frac{r}{p+q}+\frac{p}{q+r}+\frac{q}{p+r} ?
$$
Answer: 42.

Solution. Multiply the two given equalities, we get

$$
5 \cdot 9=\frac{p+q+r}{p+q}+\frac{p+q+r}{q+r}+\frac{p+q+r}{p+r}=\left(1+\frac{r}{p+q}\right)+\left(1+\frac{p}{q+r}\right)+\left(1+\frac{q}{p+r}\right)
$$

Subtract 3 from both sides of the equation, we get

$$
\frac{r}{p+q}+\frac{p}{q+r}+\frac{q}{p+r}=42
$$