**Question 1**:

Let $x_{i}>0$, ($i=1,2,\cdots,n$) and such that
$$x_{1}+x_{2}+\cdots+x_{n}=\pi.$$
Show that
$$
\dfrac{\sin{x_{1}}\sin{x_{2}}\cdots\sin{x_{n}}}{\sin{(x_{1}+x_{2})}\sin{(x_{2}+x_{3})}\cdots\sin{(x_{n}+x_{1})}}\le\left(\dfrac{\sin{\frac{\pi}{n}}}{\sin{\frac{2\pi}{n}}}\right)^n
$$
**Question 2 (may not hold):**
if $f''(x)\le 0,x\in I$, can we prove the following inequality?
$$
\begin{split}
f(x_{1}+x_{2})+&f(x_{2}+x_{3})+\ldots+f(x_{n}+x_{1})+nf\left(\dfrac{x_{1}+x_{2}+\ldots+x_{n}}{n}\right)\\
&\ge f(x_{1})+f(x_{2})+\ldots+f(x_{n})+nf\left(\dfrac{2(x_{1}+x_{2}+\ldots+x_{n})}{n}\right),
\end{split}
$$
where $x_{i}\in I$, $i=1,2,3\ldots,n$. I tried everything, but failed.

As an example of Question 2, consider $f(x)=\ln{\sin{x}}$, $0<x<\pi$. Since $$f''(x)=-\csc^2{x}<0$$ it suffices to prove that $$ \begin{split} f(x_{1}+x_{2})+f(x_{2}+x_{3})+&\ldots+f(x_{n}+x_{1})+nf\Big(\dfrac{\pi}{n}\Big)\\ &\ge f(x_{1})+f(x_{2})+\ldots+f(x_{n})+nf\Big(\dfrac{2\pi}{n}\Big) \end{split} $$ or $$ \begin{split} f(x_{1}+x_{2})+&f(x_{2}+x_{3})+\ldots+f(x_{n}+x_{1})+nf\left(\dfrac{x_{1}+x_{2}+\ldots+x_{n}}{n}\right)\\ &\ge f(x_{1})+f(x_{2})+\ldots+f(x_{n})+nf\left(\dfrac{2(x_{1}+x_{2}+\ldots+x_{n})}{n}\right). \end{split}$$ In other words,if $f''(x)\le 0$, can we prove following inequality? $$ \begin{split} f(x_{1}+x_{2})+&f(x_{2}+x_{3})+\ldots+f(x_{n}+x_{1})+nf\left(\dfrac{x_{1}+x_{2}+\ldots+x_{n}}{n}\right)\\ &\ge f(x_{1})+f(x_{2})+\ldots+f(x_{n})+nf\left(\dfrac{2(x_{1}+x_{2}+\ldots+x_{n})}{n}\right)? \end{split} $$