The sets of real numbers $\left\{x_{1}, x_{2}, x_{3}, \ldots, x_{20}\right\}$, confined between 0 and 1, are considered such that $x_{1} x_{2} x_{3} \ldots x_{20}=\left(1-x_{1}\right)\left(1-x_{2}\right)\left(1-x_{3}\right) \ldots\left(1-x_{20}\right)$. Find among these sets the one for which the value of $x_{1} x_{2} x_{3} \ldots x_{20}$ is maximal.
The value of the product $x_{1} x_{2} \ldots x_{20}$ is maximal at the same time as the value of the expression $\left(x_{1} x_{2} \ldots x_{20}\right)^{2}=$ $x_{1} x_{2} \ldots x_{20}\left(1-x_{1}\right)\left(1-x_{2}\right) \ldots\left(1-x_{20}\right)$. The last product is broken down into factors of the form $x_{k}\left(1-x_{k}\right)$, each of which (for example, according to the Cauchy inequality) is maximal at $x_{k}=1 / 2$.

## Answer

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