What are the conditions for the roots of the equations $a_{1} x^{2}+b_{1} x+c_{1}=0$ and $a_{2} x^{2}+b_{2} x+c_{2}=0$ to be equal?
If the roots are equal, then their sum and product are also equal, and vice versa, if the sum and product of the roots are equal, then the roots are also equal, so if

$$
\frac{b_{1}}{a_{1}}=\frac{b_{2}}{a_{2}} \quad \text { and } \quad \frac{c_{1}}{a_{1}}=\frac{c_{2}}{a_{2}}
$$

or

$$
\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}
$$

then the roots are equal.

(József Berger, Budapest.)

The problem was also solved by: Auer Gy., Bauer E., Bayer N., Csada I., Ehrenfeld N., Erdős V., Kirchknopf E., Kiss J., Kovács Gy., Kürth R., Lusztig M., Paunz A., Spitzer L., Wáhl V.