On Shapiro’s cyclic inequality for $N=13$

Abstract:

A cyclic sum $S({\mathbf {x}}) = \Sigma \;{x_i}/({x_{i + 1}} + {x_{i + 2}})$ is formed with the N components of a vector x, where ${x_{N + 1}} = {x_1}$, ${x_{N + 2}} = {x_2}$, and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors x for which $S({\mathbf {x}}) < N/2$ if $N \geqslant 14$ and even, and if $N \geqslant 25$. It has been proved that the inequality $S({\mathbf {x}}) \geqslant N/2$ holds for $N \leqslant 12$. Although it has been conjectured repeatedly that the inequality also holds for odd N between 13 and 23. this has apparently not yet been proved. Here we will confirm that the inequality indeed holds for $N = 13$.