Question 72
Determine all triples $(a, b, c)$ of real numbers for which the following relations hold simultaneously:
$$
\begin{aligned}
& a\left(b^{2}+c\right)=c(c+a b) \\
& b\left(c^{2}+a\right)=a(a+b c) \\
& c\left(a^{2}+b\right)=b(b+c a)
\end{aligned}
$$
$$
\begin{aligned}
& a\left(b^{2}+c\right)=c(c+a b) \\
& b\left(c^{2}+a\right)=a(a+b c) \\
& c\left(a^{2}+b\right)=b(b+c a)
\end{aligned}
$$