for $n \ge 4$ but I do not know how to solve $n=1,2,3.$ Sorry, I am still developing my inequality skills. Edit: thanks @below. Only $n=3$ left to go.

For $n=1$ we show that: $$\sum_{\text{cyc}}\sqrt x\ge\sum_{\text{cyc}}xy$$which is true since $\sqrt x-\frac12xy-\frac12xz=\frac12\left(\sqrt x-1\right)^2\left(x+2\sqrt x\right)\ge0$ and summing cyclically. The case $n=2$ expands to the same thing.