Shapiro inequality

Let $n$ be a positive integer and let $x_{1},\ldots,x_{n}$ be positive reals. If $n$ is even and $n\leq 12$ , or $n$ is odd and $n\leq 23$ , then

\sum_{i=1}^{n}\frac{x_{i}}{x_{i+1}+x_{i+2}}\geq\frac{n}{2},

where the subscripts are to be understood modulo $n$ .

The particular case of $n=3$ is also known as Nesbitts inequality.