<s>[INST] Find the largest $t$ such that for all positive $x, y, z$ the following inequality is satisfied Find the largest $t$ such that for all positive $x, y, z$ the following inequality is satisfied:
$(xy+xz+yz) \left(\dfrac{1}{x+y}+\dfrac{1}{x+z}+\dfrac{1}{y+z}\right)^2 \geq t$.
If there were such an inequality:
$ t_{1} \leq (xy+xz+yz) \left(\dfrac{1}{x+y}+\dfrac{1}{x+z}+\dfrac{1}{y+z}\right)^2 \leq t_{2}$, it is obvious that $t_{1}=0$ is suitable. And if $x=y=z$, then we have:
$(xy+xz+yz) \left(\dfrac{1}{x+y}+\dfrac{1}{x+z}+\dfrac{1}{y+z}\right)^2=3x^2 \cdot \dfrac{9}{4x^2}=\dfrac{27}{4}$. Next question is this value is the maximum or minimum?
If $x=1, y=1, z=0$, then we have $F(1,1,0)=(\dfrac{1}{2}+2)^2=\dfrac{25}{4}< \dfrac{27}{4}$, but if $x=1, y=1, z=n$, then we have $F(1,1,n)=(1+2n)(\dfrac{1}{2}+\dfrac{2}{1+n})^2>(1+2n)(\dfrac{1}{4}) \rightarrow +\infty$, which means, if I'm not mistaken, that this function has no extremum?
What to do with the case when $x,y,z$ are different I have not yet figured out.
Maybe in my case one of the following inequalities could be applied, but I'm stumped:
$1) \ xyz \geq(x+y-z)(x+z-y)(y+z-x)$
$2) \ x^3+y^3+z^3 \geq x^2y+x^2z+y^2x+y^2z+z^2x+z^2y-3xyz$
$3) \ x^3+y^3+z^3 \geq 3xyz$.
Maybe someone can give an idea where to start with my case? Thanks!!
 [/INST] Another way.
By my previous post we need to prove that:
$$\sum_{cyc}xy\left(\sum_{cyc}\frac{1}{x+y}\right)^2\geq\frac{25}{4},$$ which is true by Muirhead:
$$\sum_{cyc}xy\left(\sum_{cyc}\frac{1}{x+y}\right)^2-\frac{25}{4}=$$
$$=\frac{\sum\limits_{sym}\left(4x^5y-x^4y^2-3x^3y^3+x^4yz+14x^3y^2z+\frac{19}{3}x^2y^2z^2\right)}{4\prod\limits_{cyc}(x+y)^2}\geq0.$$
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