Inequality $\frac{{x}_{1}^{2}}{{x}_{1}^{2}+{x}_{2}{x}_{3}}+\frac{{x}_{2}^{2}}{{x}_{2}^{2}+{x}_{3}{x}_{4}}+\cdots +\frac{{x}_{n-1}^{2}}{{x}_{n-1}^{2}+{x}_{n}{x}_{1}}+\frac{{x}_{n}^{2}}{{x}_{n}^{2}+{x}_{1}{x}_{2}}\le n-1$

Show that for all $n\ge 2$

$\frac{{x}_{1}^{2}}{{x}_{1}^{2}+{x}_{2}{x}_{3}}+\frac{{x}_{2}^{2}}{{x}_{2}^{2}+{x}_{3}{x}_{4}}+\cdots +\frac{{x}_{n-1}^{2}}{{x}_{n-1}^{2}+{x}_{n}{x}_{1}}+\frac{{x}_{n}^{2}}{{x}_{n}^{2}+{x}_{1}{x}_{2}}\le n-1$

where ${x}_{i}$ are real positive numbers

I was going to use $\frac{{x}_{1}^{2}}{{x}_{1}^{2}+{x}_{2}{x}_{3}}=\frac{1}{1+\frac{{x}_{2}{x}_{3}}{{x}_{1}^{2}}}\le \frac{{x}_{1}}{2}\cdot \frac{1}{\sqrt{{x}_{2}{x}_{3}}}\le \frac{{x}_{1}}{4}(\frac{1}{{x}_{2}}+\frac{1}{{x}_{3}})$

$\frac{{x}_{1}^{2}}{{x}_{1}^{2}+{x}_{2}{x}_{3}}+\frac{{x}_{2}^{2}}{{x}_{2}^{2}+{x}_{3}{x}_{4}}+...+\frac{{x}_{n-2}^{2}}{{x}_{n-2}^{2}+{x}_{n-1}{x}_{n}}+\frac{{x}_{n-1}^{2}}{{x}_{n-1}^{2}+{x}_{n}{x}_{1}}+\frac{{x}_{n}^{2}}{{x}_{n}^{2}+{x}_{1}{x}_{2}}$

$=\frac{1}{1+\frac{{x}_{2}{x}_{3}}{{x}_{1}^{2}}}+\frac{1}{1+\frac{{x}_{3}{x}_{4}}{{x}_{2}^{2}}}+...+\frac{1}{1+\frac{{x}_{n-1}{x}_{n}}{{x}_{n-2}^{2}}}+\frac{1}{1+\frac{{x}_{n}{x}_{1}}{{x}_{n-1}^{2}}}+\frac{1}{1+\frac{{x}_{1}{x}_{2}}{{x}_{n}^{2}}}$

$\le \frac{{x}_{1}}{4}(\frac{1}{{x}_{2}}+\frac{1}{{x}_{3}})+\frac{{x}_{2}}{4}(\frac{1}{{x}_{3}}+\frac{1}{{x}_{4}})+...+\frac{{x}_{n-2}}{4}(\frac{1}{{x}_{n-1}}+\frac{1}{{x}_{n}})+\frac{{x}_{n-1}}{4}(\frac{1}{{x}_{n}}+\frac{1}{{x}_{1}})+\frac{{x}_{n}}{4}(\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}})$

$=\frac{1}{4}((\frac{{x}_{1}}{{x}_{2}}+\frac{{x}_{1}}{{x}_{3}})+(\frac{{x}_{2}}{{x}_{3}}+\frac{{x}_{2}}{{x}_{4}})+(\frac{{x}_{3}}{{x}_{4}}+\frac{{x}_{3}}{{x}_{5}})+...+(\frac{{x}_{n-2}}{{x}_{n-1}}+\frac{{x}_{n-2}}{{x}_{n}})+(\frac{{x}_{n-1}}{{x}_{n}}+\frac{{x}_{n-1}}{{x}_{1}})+(\frac{{x}_{n}}{{x}_{1}}+\frac{{x}_{n}}{{x}_{2}}))$

$=\frac{1}{4}(\left(\frac{{x}_{1}+{x}_{2}}{{x}_{3}}\right)+\left(\frac{{x}_{2}+{x}_{3}}{{x}_{4}}\right)+\left(\frac{{x}_{3}+{x}_{4}}{{x}_{5}}\right)+...+\left(\frac{{x}_{n-3}+{x}_{n-2}}{{x}_{n-1}}\right)+\left(\frac{{x}_{n-1}+{x}_{n-2}}{{x}_{n}}\right)+\left(\frac{{x}_{1}+{x}_{n}}{{x}_{2}}\right)+\left(\frac{{x}_{n-1}+{x}_{n}}{{x}_{1}}\right))$

....I thought about using Cauchy's inequality, but that would only increase the problem

Show that for all $n\ge 2$

$\frac{{x}_{1}^{2}}{{x}_{1}^{2}+{x}_{2}{x}_{3}}+\frac{{x}_{2}^{2}}{{x}_{2}^{2}+{x}_{3}{x}_{4}}+\cdots +\frac{{x}_{n-1}^{2}}{{x}_{n-1}^{2}+{x}_{n}{x}_{1}}+\frac{{x}_{n}^{2}}{{x}_{n}^{2}+{x}_{1}{x}_{2}}\le n-1$

where ${x}_{i}$ are real positive numbers

I was going to use $\frac{{x}_{1}^{2}}{{x}_{1}^{2}+{x}_{2}{x}_{3}}=\frac{1}{1+\frac{{x}_{2}{x}_{3}}{{x}_{1}^{2}}}\le \frac{{x}_{1}}{2}\cdot \frac{1}{\sqrt{{x}_{2}{x}_{3}}}\le \frac{{x}_{1}}{4}(\frac{1}{{x}_{2}}+\frac{1}{{x}_{3}})$

$\frac{{x}_{1}^{2}}{{x}_{1}^{2}+{x}_{2}{x}_{3}}+\frac{{x}_{2}^{2}}{{x}_{2}^{2}+{x}_{3}{x}_{4}}+...+\frac{{x}_{n-2}^{2}}{{x}_{n-2}^{2}+{x}_{n-1}{x}_{n}}+\frac{{x}_{n-1}^{2}}{{x}_{n-1}^{2}+{x}_{n}{x}_{1}}+\frac{{x}_{n}^{2}}{{x}_{n}^{2}+{x}_{1}{x}_{2}}$

$=\frac{1}{1+\frac{{x}_{2}{x}_{3}}{{x}_{1}^{2}}}+\frac{1}{1+\frac{{x}_{3}{x}_{4}}{{x}_{2}^{2}}}+...+\frac{1}{1+\frac{{x}_{n-1}{x}_{n}}{{x}_{n-2}^{2}}}+\frac{1}{1+\frac{{x}_{n}{x}_{1}}{{x}_{n-1}^{2}}}+\frac{1}{1+\frac{{x}_{1}{x}_{2}}{{x}_{n}^{2}}}$

$\le \frac{{x}_{1}}{4}(\frac{1}{{x}_{2}}+\frac{1}{{x}_{3}})+\frac{{x}_{2}}{4}(\frac{1}{{x}_{3}}+\frac{1}{{x}_{4}})+...+\frac{{x}_{n-2}}{4}(\frac{1}{{x}_{n-1}}+\frac{1}{{x}_{n}})+\frac{{x}_{n-1}}{4}(\frac{1}{{x}_{n}}+\frac{1}{{x}_{1}})+\frac{{x}_{n}}{4}(\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}})$

$=\frac{1}{4}((\frac{{x}_{1}}{{x}_{2}}+\frac{{x}_{1}}{{x}_{3}})+(\frac{{x}_{2}}{{x}_{3}}+\frac{{x}_{2}}{{x}_{4}})+(\frac{{x}_{3}}{{x}_{4}}+\frac{{x}_{3}}{{x}_{5}})+...+(\frac{{x}_{n-2}}{{x}_{n-1}}+\frac{{x}_{n-2}}{{x}_{n}})+(\frac{{x}_{n-1}}{{x}_{n}}+\frac{{x}_{n-1}}{{x}_{1}})+(\frac{{x}_{n}}{{x}_{1}}+\frac{{x}_{n}}{{x}_{2}}))$

$=\frac{1}{4}(\left(\frac{{x}_{1}+{x}_{2}}{{x}_{3}}\right)+\left(\frac{{x}_{2}+{x}_{3}}{{x}_{4}}\right)+\left(\frac{{x}_{3}+{x}_{4}}{{x}_{5}}\right)+...+\left(\frac{{x}_{n-3}+{x}_{n-2}}{{x}_{n-1}}\right)+\left(\frac{{x}_{n-1}+{x}_{n-2}}{{x}_{n}}\right)+\left(\frac{{x}_{1}+{x}_{n}}{{x}_{2}}\right)+\left(\frac{{x}_{n-1}+{x}_{n}}{{x}_{1}}\right))$

....I thought about using Cauchy's inequality, but that would only increase the problem