The sum of non-negative numbers $x_{1}, x_{2}, \ldots, x_{10}$ is 1. Find the maximum possible value of the sum $x_{1} x_{2}+$ $x_{2} x_{3}+\ldots+x_{9} x_{10}$
Let's use the well-known inequality $a b \leq 1 / 4(a+b)^{2}$.

$x_{1} x_{2}+x_{2} x_{3}+\ldots+x_{9} x_{10} \leq\left(x_{1}+x_{3}+\ldots+x_{9}\right)\left(x_{2}+x_{4}+\ldots+x_{10}\right) \leq 1 / 4\left(x_{1}+x_{2}+\ldots+x_{10}\right)^{2}=1 / 4$.

Equality, for example, is achieved if $x_{1}=x_{2}=1 / 2, x_{3}=x_{4}=\ldots=x_{10}=0$.

## Answer

$0.25$.