Let's calculate the expression

$$
\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}
$$

if $x_{1}$ and $x_{2}$ are the roots of the equation

$$
x^{2}+p x+q=0
$$
$$
\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}=\frac{x_{1}^{2}+x_{2}^{2}}{x_{1} x_{2}}=\frac{\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}}{x_{1} x_{2}}
$$

Given

$$
x_{1}+x_{2}=-p \text { and } x_{1} x_{2}=q
$$

thus

$$
\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}=\frac{p^{2}-2 q}{q}
$$

(Viola Rezsó, Budapest, V. district.)

The problem was also solved by: Babocsai G., Bayer N., Bokor K., Chambré M., Czukor G., Ehrenstein I., Erdős V., Fried E., Grossberger Z., Havas B., Kirchknopf E., Koffer B., Kovács Gy., Lázár B., Miklóssy K., Neumann L., Paunz A., Petrik S., Rosenthal M., Sárközy E., Szilárd V., Tandlich E., Tóth B., Weisz Á.