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I'm proud to present the work of my last week end and I think it's hard to prove :

Let $x,y,z>0$ such that $xyz=1$ then we have : $$\frac{x^{x^2}}{x^2+y^2}+\frac{y^{y^2}}{y^2+z^2}+\frac{z^{z^2}}{z^2+x^2}\geq \frac{3}{2}$$

I try the following inequality to prove my statement : $$\frac{x^{x^2}}{x^2+y^2}+\frac{y^{y^2}}{y^2+z^2}+\frac{z^{z^2}}{z^2+x^2}\geq \frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+x}\geq \frac{3}{2}$$

But it doesn't work .

I try also to apply the well-know inequality :$e^x\geq x+1$ but it gives nothing consequent. So I'm a bit lost with this inequality ...

Any help is appeciated . Thanks

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The hint.

Prove that: $$x^{x^2}\geq 2x^3-4x^2+3x.$$ The rest is smooth.

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