<s>[INST] If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $ 
Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that
  $$
\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n}
$$
  for each integer $n$.

I have used Cauchy-Schwarz inequalities and Jensen inequality. But I am stuck. I need some idea and advice on this problem. Induction would be cruel. 
 [/INST] You can see that every term has value $\frac{1}{n}$ in case $a = b = c = 1$.
Try to prove that changing one of the variables (let $a > 1$,  $b < 1$,  $c < 1$ be the first case and $a > 1$, $b > 1$, $c < 1$ - the second one and $a > 1$, $b < 1$, $c > 1$ - the third one) increases the total sum.
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