What relationships must the coefficients of the expression

$$
a x^{2}+b y^{2}+c z^{2}+2 d x y+2 e y z+2 f z x
$$

satisfy so that this expression is a perfect square?
For the above expression to be equal to

$$
(x \sqrt{a}+y \sqrt{b}+z \sqrt{c})^{2},
$$

it is necessary that

$$
\sqrt{a b}=d, \quad \sqrt{b c}=e, \quad \sqrt{c a}=f
$$

which means

$$
a b-d^{2}=b c-e^{2}=c a-f^{2}=0
$$

which can also be replaced by the following relations:

$$
a e-d f=b f-d e=c d-e f=0
$$

The problem was solved by: Friedmann Bernát, fg. VI. S. Ujhely; Suták Sándor fg. VIII. Nyíregyháza.