[ $\underline{\text { Classical Inequalities (Miscellaneous) })]}$

$$
\sqrt{x_{1}^{2}+\left(1-x_{2}\right)^{2}}+\sqrt{x_{2}^{2}+\left(1-x_{3}\right)^{2}}+\ldots+\sqrt{x_{2 n}^{2}+\left(1-x_{1}\right)^{2}}
$$
According to the inequality between the quadratic mean and the arithmetic mean

$$
\begin{aligned}
& \sqrt{2}\left(\sqrt{x_{1}^{2}+\left(1-x_{2}\right)^{2}}+\sqrt{x_{2}^{2}+\left(1-x_{3}\right)^{2}}+\ldots+\sqrt{x_{n}^{2}+\left(1-x_{1}\right)^{2}}\right) \geq \\
& \geq\left|x_{1}\right|+\left|1-x_{2}\right|+\left|x_{2}\right|+\left|1-x_{3}\right|+\ldots+\left|x_{n}\right|+\left|1-x_{1}\right| \geq x_{1}+1-x_{2}+x_{2}+1-x_{3}+\ldots+x_{n}+1-x_{1}=n
\end{aligned}
$$

Equality is achieved when $x_{1}=x_{2}=\ldots=x_{n}=1 / 2$.

## Answer

$\frac{n}{\sqrt{2}}$.