Let $f(x\_1,\ldots , x\_n) = \frac{x\_1}{x\_2+x\_3} + \frac{x\_2}{x\_3+x\_4} + \cdots + \frac{x\_n}{x\_1+x\_2}$, defined for $x\_i>0$.


1. Is there $(x\_1, \ldots ,x\_n)\in {\mathbb{R}^\*\_+}^n$ such that $f(x\_1,\ldots , x\_n) < n/2$?
2. Can we find $\inf\_{x\_i>0}f(x\_1,\ldots , x\_n)$?



Inf of a mutivariate function
This is discussed briefly as a generalization of Shapiro's cyclic sum
inequality by J. Michael Steele in his book
[The Cauchy-Schwarz Master Class](http://books.google.co.uk/books?id=7Fm3r9jcbqYC&lpg=PP1&dq=Steele%2520Cauchy&pg=PA104#v=onepage&q=Shapiro%27s&f=false).
He remarks that (1.) holds for $n\ge25$ and refers to this paper:
P. J. Bushell, Shapiro’s “Cyclic Sums", 
*Bull. L.M.S.* (1994) **26**, 564–574.


