<s>[INST] how to continue after this substitution? 
Given $abc=1$, prove that $${\frac{a}{a^2+2}}+ {\frac{b}{b^2+2}}+{\frac{c}{c^2+2}} \leqslant 1 $$

I tried substitution  $a={\frac{1}{x}},b={\frac{1}{y}},c={\frac{1}{z}}$ but can't finish.
 [/INST] Let $a=\frac{x}{y},b=\frac{y}{z},c=\frac{z}{x}$ we have to prove

$$\sum_{cyc}\frac{xy}{x^2+2y^2}\le 1$$ Now as $x^2+y^2\ge 2xy$ it suffices to prove


$$\sum_{cyc}\frac{x}{2x+y}\le 1 $$ $$\iff \sum_{cyc} \frac{y}{2x+y}\ge 1$$ which is true as


$$\sum_{cyc} \frac{y}{2x+y}= \sum_{cyc} \frac{y^2}{2xy+y^2}\ge \frac{{(x+y+z)}^2}{2xy+y^2+2zx+x^2+2yz+z^2}=1$$

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