Peryn A.

Solve the system of equations in positive numbers

$$
\left\{\begin{array}{l}
x_{1}+\frac{1}{x_{2}}=4 \\
x_{2}+\frac{1}{x_{3}}=1 \\
x_{3}+\frac{1}{x_{1}}=4 \\
\ldots \ldots \ldots \ldots \\
x_{99}+\frac{1}{x_{100}}=4 \\
x_{100}+\frac{1}{x_{1}}=1
\end{array}\right.
$$
Due to the inequality between the arithmetic mean and the geometric mean

$$
x_{1}+\frac{1}{x_{2}} \geq 2 \sqrt{\frac{x_{1}}{x_{2}}}, x_{2}+\frac{1}{x_{3}} \geq 2 \sqrt{\frac{x_{2}}{x_{3}}}, \ldots, x_{100}+\frac{1}{x_{1}} \geq 2 \sqrt{\frac{x_{100}}{x_{1}}}
$$

Multiplying these inequalities, we obtain the inequality $\left(x_{1}+\frac{1}{x_{2}}\right)\left(x_{2}+\frac{1}{x_{3}}\right) \ldots\left(x_{100}+\frac{1}{x_{1}}\right) \geq 2^{100}=4^{50}$. Multiplying the equations of the system, we see that this inequality becomes an equality. Therefore, each of the inequalities (*) must become an equality, that is, $x_{1}=1 / x_{2}$,

$x_{2}=1 / x_{3}, \ldots, x_{100}=1 / x_{1}$. Substituting the obtained expressions for $x_{1}, x_{2}, \ldots, x_{100}$ into the given system, we find the answer.

## Answer

$(2,1 / 2,2,1 / 2, \ldots, 2,1 / 2)$