This is my problem:
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 used the Cauchy inequality but this is the closest I got: $$\frac{a^{2n}}{a^n + (n-1)\,b^n} + \frac{b^{2n}}{b^n + (n-1)\,c^n} + \frac{c^{2n}}{c^n + (n-1)\,a^n} \geq \frac{3}{n}$$ I get the same with Jensen’s inequality. If I can show it for $2n$, how do I show it for $n+2$?