<s>[INST] AM-GM inequality On the wikipedia page on 
"Nesbit's inequality", the fifth proof ends as follows:
$$ \frac{x+z}{y}+\frac{y+z}{x}+\frac{x+y}{z}\geq 6$$
which is true, by AM-GM inequality.
I am wondering if the inequality is obvious / immediate from just looking at it and how you see this immediately without resorting to something like the following proof:
\begin{align*}
\frac{x+z}{y}+\frac{y+z}{x}+\frac{x+y}{z} &\geq 
2\cdot\left(\frac{\sqrt{xz}}{y}+\frac{\sqrt{yz}}{x}+\frac{\sqrt{xy}}{z}\right) \\
&\geq 
6\cdot\left(\frac{\sqrt{xz}}{y} \cdot \frac{\sqrt{yz}}{x} \cdot\frac{\sqrt{xy}}{z}\right)^{\frac13} \\
&=6\cdot(1)^{\frac13}=6
\end{align*}
 [/INST] $$ \frac{x+z}{y}+\frac{y+z}{x}+\frac{x+y}{z}\geq 6$$ is equivalent to
$$ \frac{x^2 z+z^2x + y^2 z+z^2 y + x^2 y + y^2 x}{6}\geq xyz$$
which is true by the AM-GM inequality:
$$ \frac{x^2 z+z^2x + y^2 z+z^2 y + x^2 y + y^2 x}{6}\geq \sqrt[6]{x^6y^6z^6}.$$
</s>