Question 73
Suppose, for some pair of real numbers $x, y$, the real numbers $a, b, c$ satisfy the five equations
$$
\begin{aligned}
a+b+c &= 1 & bc-x-2ya &= 0 \\
a^2+b^2+c^2 &= 3 & ca-x-2yb &= 0 \\
ab-x-2yc &= 0.
\end{aligned}
$$
Prove that $abc$ is either $-1$ or $\frac{5}{27}$.
$$
\begin{aligned}
a+b+c &= 1 & bc-x-2ya &= 0 \\
a^2+b^2+c^2 &= 3 & ca-x-2yb &= 0 \\
ab-x-2yc &= 0.
\end{aligned}
$$
Prove that $abc$ is either $-1$ or $\frac{5}{27}$.