Let's modify the equation whose roots are the reciprocals of the roots of the equation $x^{2}+p x+q=0$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly. 

---

Let's modify the equation whose roots are the reciprocals of the roots of the equation $x^{2}+p x+q=0$.
If the roots of the given equation are $x_{1}$ and $x_{2}$, and $x^{2}+r x+s=0$ is the equation to be found, then

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

and

$$
\frac{1}{x_{1}} \cdot \frac{1}{x_{2}}=\frac{1}{q}=s
$$

thus

$$
x^{2}+\frac{p}{q} x+\frac{1}{q}=0
$$

or

$$
q x^{2}+p x+1=0
$$

is the equation to be found.

(Nándor Bayer, Losonc.)

The problem was also solved by: Auer Gy., Bauer E., Csada I., Dénes M., Ehrenfeld N., Engler J., Erdős V., Fried E., Kirchknopf E., Kiss J., Kovács Gy., Lusztig M., Paunz A., Rosenthal M., Sárközy P., Spitzer L., Szőke D., Wáhl V.