Solve the following equation:

$$
\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0
$$
Our equation can also be written as:

$$
(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0
$$

or

$$
3 x^{2}-2 x(a+b+c)+(a b+b c+c a)=0
$$

from which

$$
x=\frac{a+b+c \pm \sqrt{(a+b+c)^{2}-3(a b+b c+c a)}}{3}
$$

(Hay Andor, Pécs.)