Let the roots of the equation $\operatorname{Az} x^{2}-m x+n=0$ be $\tan \varphi$ and $\tan \phi$. Determine the angles $\varphi$ and $\phi$ without solving the equation.
Given $\operatorname{tg} \varphi+\operatorname{tg} \phi=m$ and $\operatorname{tg} \varphi \cdot \operatorname{tg} \phi=n$, therefore

$$
\operatorname{tg}(\varphi+\phi)=\frac{\operatorname{tg} \varphi+\operatorname{tg} \phi}{1-\operatorname{tg} \varphi \operatorname{tg} \phi}=\frac{m}{1-n}
$$

Furthermore,

$$
\operatorname{tg}^{2}(\varphi-\phi)=\left(\frac{\operatorname{tg} \varphi-\operatorname{tg} \phi}{1+\operatorname{tg} \varphi \operatorname{tg} \phi}\right)^{2}=\frac{(\operatorname{tg} \varphi+\operatorname{tg} \phi)^{2}-4 \operatorname{tg} \varphi \cdot \operatorname{tg} \phi}{(1+\operatorname{tg} \varphi \operatorname{tg} \phi)^{2}}=\frac{m^{2}-4 n}{(1+n)^{2}}
$$

thus

$$
\operatorname{tg}(\varphi-\phi)=\frac{\sqrt{m^{2}-4 n}}{1+n}
$$

(1) and (2) determine $\operatorname{tg} \varphi$ and $\operatorname{tg} \phi$.

(Alfréd Haar, Budapest.)

The problem was also solved by: Bartók I., Deutsch I., Enyedi B., Hirschfeld Gy., Kertész G., Klein A., König D., Ligeti P., Pivnyik I., Raab R., Schmidl I., Schwarz Gy., Tóbiás L., Wohlstein S.