Solve the following system of equations:

$$
\begin{aligned}
x-a y+a^{2} z & =a^{3} \\
x-b y+b^{2} z & =b^{3} \\
x-c y+c^{2} z & =c^{3} .
\end{aligned}
$$
From (1) subtract (2) and (3):

$$
(b-a) y-(b-a)(b+a) z=-(b-a)\left(b^{2}+a b+a^{2}\right)
$$

$$
(c-a) y-(c-a)(c+a) z=-(c-a)\left(c^{2}+a c+a^{2}\right)
$$

If $a$ is not equal to $b$ and $c$ is not equal to $a$, then both sides of (4) and (5) are divisible by $(b-a)$ and $(c-a)$, respectively; after this division, subtract (5) from (4) and rearrange:

$$
z=a+b+c
$$

Substituting this value of $z$ into (4) or (5):

$$
y=a b+a c+b c
$$

and finally from (1):

$$
x=a b c
$$

(Jenö Scharff, Budapest.)

The problem was also solved by: Aczél F., Baumann J., Bayer B., Burján K., Czank K., Demeter J., Demjén E., Faith F., Fekete N., Filkorn J., Freudenberg K., Goldstein Á., Hein J., Hirschfeld Gy., Hoffmann M., Izsáky L., Jankovich S., Keesz J., Kende B., Kerekes T., Kertész G., Kőnig D., Krausz B., Krausz F., Krisztián Gy., Kürth A., Lichtig A., Lukhaub Gy., Lupsa Gy., Messik G., Messik V., Mikuleczky I., Moskovits Zs., Papp F., Perl Gy., Perlesz D., Póka Gy., Reich Zs., Rosenberg Á., Salczmann S., Sasvári G., Sasvári J., Schor J., Selényi M., Singer A., Smodics K., Spitzer H., Stern D., Stromfeld F., Tézner E., Tóth S., Weisz A., Weisz J., Winter F.