Solve the following equation:

$$
x+\sqrt[3]{a-x^{3}}=\frac{c}{x \sqrt[3]{a-x^{3}}}
$$
The given equation can also be written as:

$$
x \sqrt[3]{a-x^{3}}\left(x+\sqrt[3]{\left(a-x^{3}\right)}=c\right.
$$

or

$$
x^{2} \sqrt[3]{a-x^{3}}+x \sqrt[3]{\left(a-x^{3}\right)^{2}}=c
$$

Cubing both sides of (1):

$$
\left[x^{3}+3\left(x^{2} \sqrt[3]{a-x^{3}}+x \sqrt[3]{\left(a-x^{3}\right)^{2}}+a-x^{3}\right) x^{3}\left(a-x^{3}\right)=c^{3}\right.
$$

Considering (2), we get:

$$
(3 c+a) x^{3}\left(a-x^{3}\right)=c^{3}
$$

Considering (2), we get:

$$
(3 c+a) x^{3}\left(a-x^{3}\right)=c^{3}
$$

or

$$
x^{6}-a x^{3}=-\frac{c^{3}}{3 c+a}
$$

from which

$$
x=\sqrt[3]{\frac{a}{2} \pm \sqrt{\frac{a^{2}}{4}-\frac{c^{3}}{3 c+a}}}
$$

(Mihály Devecis.)

The problem was also solved by: Barna D., Dénes A., Détshy K., Erdős A., Fekete J., Freibauer E., Goldziher K., Kármán T., Kertész L., Manheim E., Schwartz E., Spitzer Ö., Szabó I., Szabó K., Tüske J., Weisz Á., Weisz J.