<s>[INST] If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$? If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$?
I tried substitution on the bottom (for example $\frac{b-1}{\frac{1}{a}+1}$), but I then a very similar term. What should I do?
I can not find a way to decompose of combine.
Also, how do you prove that $x^4+y^4+z^2 \ge xyz \sqrt{8}$?
I got $x^4+y^4+z^2\geq3\sqrt[3]{x^4*y^4*z^2}=3xy\sqrt[3]{xyz^2}.$
I do not know what to do from here.
 [/INST] For the first inequality, using the substitution of $ a = \frac{y}{x}, b = \frac{z}{y},c = \frac{x}{z}$, (which is a good approach to nomarlize these inequalities), we WTS   (see Michaels' solution)
$$  \sum y \times \frac{1}{ (x+y+z) - y } \geq \sum x \times \frac{ 1}{(x+y+z) - y }.$$
This is just rearrangement inequality, since $\{ y\}$ and $\{ \frac{ 1}{ (x+y+z) - y } \}$ are similarly ordered.
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