Question 105
Determine all triples of real numbers $(a, b, c)$ that satisfy simultaneously the equations:
$$
\begin{align*}
a(b^2 + c) &= c(c + ab), \\
b(c^2 + a) &= a(a + bc), \\
c(a^2 + b) &= b(b + ca).
\end{align*}
$$
$$
\begin{align*}
a(b^2 + c) &= c(c + ab), \\
b(c^2 + a) &= a(a + bc), \\
c(a^2 + b) &= b(b + ca).
\end{align*}
$$