What relationship must hold between $p$ and $q$ so that the roots of the equation satisfy the following condition:

$$
x_{1}^{2}+x_{2}^{2}=1
$$
Minthogy

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

therefore the

$$
x_{1}^{2}+x_{2}^{2}=1
$$

equality holds if

$$
p^{2}-2 q=1
$$

(Andor Vilcsek, Eger.)

The problem was solved by: Bauer E., Böhm J., Breuer P., Czigler I., Dénes M., Döri V., Engler J., Erdős V., Frankl F., Fried E., Gianone O., Grün E., Grünvald Gy., Koffler B., Köhler I., Lendvai D., Neumann L., Sichermann F., Spitzer L., Szántó L., Szilárd V., Viola R., Weinberger Gy.