## [ Systems of nonlinear algebraic equations ]

[ Completing the square. Sums of squares ]

Find all real solutions of the system of equations $\frac{2 x_{1}^{2}}{1+x_{1}^{2}}=x_{2}, \frac{2 x_{2}^{2}}{1+x_{2}^{2}}=x_{3}, \frac{2 x_{3}^{2}}{1+x_{3}^{2}}=x_{1}$.
If one of the numbers $x_{1}, x_{2}, x_{3}$ is zero, then the other numbers are also zero. Therefore, we will further assume that $x_{1} x_{2} x_{3} \neq 0$. Then the system can be written as $\frac{2}{x_{2}}=1+\frac{1}{x_{1}^{2}}, \frac{2}{x_{3}}=1+\frac{1}{x_{2}^{2}}, \frac{2}{x_{1}}=1+\frac{1}{x_{3}^{2}}$. Adding these equations, we get $\left(1-1 / x_{1}\right)^{2}+\left(1-\frac{1}{x_{2}}\right)^{2}+\left(1-1 / x_{3}\right)^{2}=0$. This equality can only hold if $1-1 / x_{1}=1-1 / x_{2}=1-1 / x_{3}=0$.

## Answer

$(0,0,0)$ and $(1,1,1)$.