[Systems of nonlinear algebraic equations]

Solve the system of equations:

$$
\begin{aligned}
& 1 / x=y+z \\
& 1 / y=z+x \\
& 1 / z=x+y
\end{aligned}
$$
The first method. Subtracting the second equation from the first, we get: $1 / x - 1 / y = y - x \leftrightarrow (y - x)(1 / xy - 1) = 0 \Leftrightarrow x = y$ or $xy = 1$. Let's consider two cases.

1) $x = y$. Eliminating the variable $y$ from the first and third equations, we get $1 / x = x + z, 1 / z = 2x \leftrightarrow z = 1 / 2x$, $1 / 2x = x \Leftrightarrow x = z = \pm \frac{1}{\sqrt{2}}$.
2) $xy = 1$. Then from the first equation, it follows that $z = 0$, and the expression $1 / z$ is undefined. Thus, this case is impossible.

The second method. Notice that $1 / x + x = 1 / y + y = 1 / z + z = x + y + z$. Let $x + y + z = A$, then $x, y$, and $z$ are the roots of the equation $1 / t + t = A$, which is equivalent to the quadratic equation $t^2 - At + 1 = 0$. Since a quadratic equation has no more than two roots, the values of some two variables must be the same. Further reasoning is similar to that considered in point 1) of the first method.

## Answer

$\left(\frac{\sqrt{2}}{2} ; \frac{\sqrt{2}}{2} ; \frac{\sqrt{2}}{2}\right) ;\left(-\frac{\sqrt{2}}{2} ;-\frac{\sqrt{2}}{2} ;-\frac{\sqrt{2}}{2}\right)$.

## Problem