In triangle $ABC$, the altitude from vertex $A$ is the harmonic mean of the two segments into which this segment divides side $BC$. What is $\operatorname{tg} \beta + \operatorname{tg} \gamma$ (where $\beta$ and $\gamma$ are the angles of the triangle)?
According to the conditions of the problem, the length of the altitude drawn from vertex $A$ is $\frac{2 p q}{p+q}$, where $p$ and $q$ are the two segments into which the altitude divides the side $B C$. It also follows from the condition that $\beta$ and $\gamma$ are acute angles; therefore,

$$
\operatorname{tg} \beta+\operatorname{tg} \gamma=\frac{\frac{2 p q}{p+q}}{p}+\frac{\frac{2 p q}{p+q}}{q}=2
$$