Shapiro inequality

Let $n$ be a positive integer and let $x_1,\mathrm{\dots },x_n$ be positive reals.If $n$ is even and $n\le 12$, or $n$ is odd and $n\le 23$, then

$$\sum _{i=1}^n\frac{x_i}{x_{i+1}+x_{i+2}}\ge \frac{n}{2},$$

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

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