[ Quadratic equations and systems of equations] Principle of the extreme (further). ]

Find all positive solutions of the system of equations

$$
\left\{\begin{array}{l}
x_{1}+x_{2}=x_{3}^{2} \\
x_{2}+x_{3}=x_{4}^{2} \\
x_{3}+x_{4}=x_{5}^{2} \\
x_{4}+x_{5}=x_{1}^{2} \\
x_{5}+x_{1}=x_{2}^{2}
\end{array}\right.
$$
Let $a$ be the largest of the numbers $x_{1}, \ldots, x_{5}$, and $b$ be the smallest of these numbers. Then $a^{2} \leq 2 a$ and $b^{2} \geq 2 b$. By the condition, the numbers $a$ and $b$ are positive, so $a \leq 2$ and $b \geq 2$, therefore, $a=b=2$.

## Answer

$(2,2,2,2,2)$.