Question 77
Recall that
$$
\begin{aligned}
(a+b+c)^{3} = a^{3} + b^{3} + c^{3} + 3 a^{2} b + 3 a b^{2} + 3 a c^{2} + 3 a^{2} c + 3 b^{2} c + 3 b c^{2} + 6 a b c
\end{aligned}
$$
Find the solutions of the system
$$
\begin{aligned}
a^{3} + 3 a b^{2} + 3 a c^{2} - 6 a b c & = 1 \\
b^{3} + 3 b a^{2} + 3 b c^{2} - 6 a b c & = 1 \\
c^{3} + 3 c a^{2} + 3 c b^{2} - 6 a b c & = 1
\end{aligned}
$$
$$
\begin{aligned}
(a+b+c)^{3} = a^{3} + b^{3} + c^{3} + 3 a^{2} b + 3 a b^{2} + 3 a c^{2} + 3 a^{2} c + 3 b^{2} c + 3 b c^{2} + 6 a b c
\end{aligned}
$$
Find the solutions of the system
$$
\begin{aligned}
a^{3} + 3 a b^{2} + 3 a c^{2} - 6 a b c & = 1 \\
b^{3} + 3 b a^{2} + 3 b c^{2} - 6 a b c & = 1 \\
c^{3} + 3 c a^{2} + 3 c b^{2} - 6 a b c & = 1
\end{aligned}
$$