Let integers $n>m>3$ be fixed and define $$S(x_1,...,x_n)=\sum_{k=1}^n\frac{x_k}{x_k+x_{k+1}+...+x_{k+m}},$$ where if the index $k+m$ exceeds $n$ then we define $x_{k+m}=x_{k+m-n}$. I am asked to give the infimum and supremum of the sum $S(x_1,...,x_n)$ in terms of $m$ and $n$ when $x_k$ runs in the set of all positive reals.
I have made some attempts on some special case of this problem, for example when the sum is cyclic for some $m$ and $n$. I have tried using AM-HM in the general case but the best I could get is a lower bound of the sum $$\sum \frac{x_1+...+\hat{x}_k+\hat{x}_{k+1}+...+\hat{x}_{k+m}+...+x_n}{x_1+x_2+...+x_n}$$ where the hat means removing the term from the sum. I have also tried to apply the general power mean inequality.
For which $m,n$ the sum over the positive reals have an infimum or supremum? And what is it in terms of $m$ and $n$?