Question 70
For any real number $\alpha$, define
$$
\operatorname{sign}(\alpha)= \begin{cases}+1 & \text{ if } \alpha\gt 0 \\ 0 & \text{ if } \alpha=0 \\ -1 & \text{ if } \alpha\lt 0\end{cases}
$$
How many triples $(x, y, z) \in \mathbb{R}^3$ satisfy the following system of equations
$$
\begin{aligned}
& x=2018-2019 \cdot \operatorname{sign}(y+z) \\
& y=2018-2019 \cdot \operatorname{sign}(z+x) \\
& z=2018-2019 \cdot \operatorname{sign}(x+y) ?
\end{aligned}
$$
$$
\operatorname{sign}(\alpha)= \begin{cases}+1 & \text{ if } \alpha\gt 0 \\ 0 & \text{ if } \alpha=0 \\ -1 & \text{ if } \alpha\lt 0\end{cases}
$$
How many triples $(x, y, z) \in \mathbb{R}^3$ satisfy the following system of equations
$$
\begin{aligned}
& x=2018-2019 \cdot \operatorname{sign}(y+z) \\
& y=2018-2019 \cdot \operatorname{sign}(z+x) \\
& z=2018-2019 \cdot \operatorname{sign}(x+y) ?
\end{aligned}
$$