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I've came across this problem which seems to be a classic mean inequality problem, but I can't get it right.

Let $x$, $y$ and $z$ be pozitive real numbers with $xyz=1$. Prove that $$\frac{x}{1+x^4} + \frac{y}{1+y^4} + \frac{z}{1+z^4} \leq \frac{3}{2}$$

My first try was $$\sqrt{1 * x^2} \leq \sqrt{\frac{1+x^4}{2}}$$ $$2x^2 \leq 1+x^4$$ $$\frac{x}{1+x^4} \leq \frac{x}{2x^2} = \frac{1}{2x}$$ then, using geometric mean and arithmetic mean, we obtain the inequality inside out.

A hint would be really apreciated.

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    if such a condition $$xyz=1$$ is given one can set $$x=a/b,y=b/c,z=c/a$$ – 2017-02-18
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    from where does it come? – 2017-02-18
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    @Dr.SonnhardGraubner I've found it in an old book with preparing problems for contests – 2017-02-18
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    do you have a link of this book? – 2017-02-18

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