Given that $x_1x_2...x_n=Q_n$，and $x_i\geq 1$ for all $i=1,...,n$, write $x_{n+1}=x_1$. Prove that for $n>2$, $$\sum_{i=1}^nx_ix_{i+1}-\sum_{i=1}^nx_i\geq \frac{n}{2}(Q_2-1)$$
I used AM-GM, and even tried using $x_i=\log y_i$, to rewrite the inequality as $$\log\Pi_{i=1}^nx_i\left(\log\Pi_{i=1}^nx_i-1\right)\geq \frac{n}{2}(\log y_1\log y_2-1)$$ But Jensen's inequality doesnt seems to apply either. Can anyone give some help. Thanks
The inequality does not hold in general. Consider for example $\,n=3\,$ and $x_1 =3, x_2=x_3=1\,$:
$$ 3 \cdot 1 + 1 \cdot 1 + 1 \cdot 3 - (3+1+1) = 2 \;\lt\; 3 = \frac{3}{2}\,(3 \cdot 1 - 1) $$