Possible values of $\frac{x_{1}}{x_{n} + x_{1} + x_{2}} + \frac{x_{2}}{x_{1} + x_{2} + x_{3}} + ... + \frac{x_{n}}{x_{n-1} + x_{n} + x_{1}}$

Question

Let $n > 3$ be fixed integer. If $x_{1}, x_{2}, x_{3}, ..., x_{n}$ are from $\mathbb{R}^{+}$, what are the possible values for

$$ \frac{x_{1}}{x_{n} + x_{1} + x_{2}} + \frac{x_{2}}{x_{1} + x_{2} + x_{3}} + ... + \frac{x_{n}}{x_{n-1} + x_{n} + x_{1}} $$

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Attempt:

Start with $n=4$, we have

$$ S = \frac{x_{1}}{x_{4} + x_{1} + x_{2}} + \frac{x_{2}}{x_{1} + x_{2} + x_{3}} + \frac{x_{3}}{x_{2} + x_{3} + x_{4}} + \frac{x_{4}}{x_{3} + x_{4} + x_{1}} $$

then extending this we get

$$ \frac{x_{1}}{x_{4} + x_{1} + x_{2}} + \frac{x_{2}}{x_{1} + x_{2} + x_{3}} + \frac{x_{3}}{x_{2} + x_{3} + x_{4}} + \frac{x_{4}}{x_{3} + x_{4} + x_{1}} $$

$$ + \frac{x_{2}}{x_{4} + x_{1} + x_{2}} + \frac{x_{3}}{x_{1} + x_{2} + x_{3}} + \frac{x_{4}}{x_{2} + x_{3} + x_{4}} + \frac{x_{1}}{x_{3} + x_{4} + x_{1}} $$

$$ + \frac{x_{4}}{x_{4} + x_{1} + x_{2}} + \frac{x_{1}}{x_{1} + x_{2} + x_{3}} + \frac{x_{2}}{x_{2} + x_{3} + x_{4}} + \frac{x_{3}}{x_{3} + x_{4} + x_{1}} $$

$$ = 4 $$

Next

$$ \frac{x_{2} + x_{4}}{x_{4} + x_{1} + x_{2}} \le \frac{x_{3} + x_{2} + x_{4}}{x_{4} + x_{1} + x_{2} + x_{3}} $$

$$ \frac{x_{1} + x_{3}}{x_{1} + x_{2} + x_{3}} \le \frac{x_{4} + x_{1} + x_{3}}{x_{4} + x_{1} + x_{2} + x_{3}} $$

$$ \frac{x_{2} + x_{4}}{x_{2} + x_{3} + x_{4}} \le \frac{x_{1} + x_{2} + x_{4}}{x_{4} + x_{1} + x_{2} + x_{3}} $$

$$ \frac{ x_{3} + x_{1} }{x_{3} + x_{4} + x_{1}} \le \frac{x_{3} + x_{2} + x_{1}}{x_{4} + x_{1} + x_{2} + x_{3}} $$

So we can get

$$ \frac{x_{1}}{x_{4} + x_{1} + x_{2}} + \frac{x_{2}}{x_{1} + x_{2} + x_{3}} + \frac{x_{3}}{x_{2} + x_{3} + x_{4}} + \frac{x_{4}}{x_{3} + x_{4} + x_{1}} $$ $$ = 4 - \left( \frac{x_{2} + x_{4}}{x_{4} + x_{1} + x_{2}} + \frac{x_{1} + x_{3}}{x_{1} + x_{2} + x_{3}} + \frac{x_{2} + x_{4}}{x_{2} + x_{3} + x_{4}} + \frac{ x_{3} + x_{1} }{x_{3} + x_{4} + x_{1}} \right) \ge 1$$

Also using similar approach we get:

$$ \frac{x_{1}}{x_{4} + x_{1} + x_{2}} + \frac{x_{2}}{x_{1} + x_{2} + x_{3}} + \frac{x_{3}}{x_{2} + x_{3} + x_{4}} + \frac{x_{4}}{x_{3} + x_{4} + x_{1}} \le \frac{x_{1} + x_{3}}{ x_{3} + x_{4} + x_{1} + x_{2}} + \frac{x_{4} +x_{2}}{x_{4} + x_{1} + x_{2} + x_{3}} + \frac{x_{1} + x_{3}}{x_{1} + x_{2} + x_{3} + x_{4}} + \frac{x_{2} + x_{4}}{x_{2} + x_{3} + x_{4} + x_{1}} = 2 $$

so far we have $1 \le S \le 2 $

Answer 1

By the same way we obtain: $$1=\sum_{cyc}\frac{x_1}{\sum\limits_{i=1}^nx_i}<\sum_{cyc}\frac{x_1}{x_n+x_1+x_2}.$$ Also, for even $n$ we obtain: $$\sum_{cyc}\frac{x_1}{x_n+x_1+x_2}<\left(\frac{x_1}{x_1+x_2}+\frac{x_2}{x_1+x_2}\right)+...+\left(\frac{x_{n-1}}{x_{n-1}+x_n}+\frac{x_n}{x_{n-1}+x_n}\right)=\frac{n}{2}.$$ For odd $n$ we can assume that $x_1+x_2+x_3$ is a minimal value of any our denominators, which gives $$\frac{x_1}{x_n+x_1+x_2}+\frac{x_2}{x_1+x_2+x_3}+\frac{x_3}{x_2+x_3+x_4}\leq$$ $$\leq\frac{x_1}{x_1+x_2+x_3}+\frac{x_2}{x_1+x_2+x_3}+\frac{x_3}{x_1+x_2+x_3}=1$$ and by the same way we obtain: $$\sum_{cyc}\frac{x_1}{x_n+x_1+x_2}<1+\frac{n-3}{2}=\frac{n-1}{2},$$ which gives $$\sum_{cyc}\frac{x_1}{x_n+x_1+x_2}<\left[\frac{n}{2}\right]$$ It's enough to understand that we got infimum and supremum.