Let $a,b,c$ be real positive numbers so that $abc=1$. Find the maximum value that the following expression can attain:
$$\frac{a}{a^8+1}+\frac{b}{b^8+1}+\frac{c}{c^8+1}$$
My try:
I first though on apply a variable change so that $a=\frac{x}{y}$, $b= \frac{y}{z}$ and $c=\frac{z}{x}$. The problem is that the problem became harder for me:
$$\sum_{cyc} \frac{xy^7}{x^8+x^7}$$
Then I though on applying Holder in the denominator of the first expression, so it would look like:
$$\sum_{cyc} \frac{a}{a^8+1} \leq \sum_{cyc} \frac{2^7a}{(a+1)^8}$$
After that, I tried applying $a+1 \geq 2\sqrt{a}$. But the expression wasn't correct anymore.