<s>[INST] How to prove this high-degree inequality Let $x$,$y$,$z$ be positive real numbers which satisfy $xyz=1$. Prove that:
$(x^{10}+y^{10}+z^{10})^2 \geq 3(x^{13}+y^{13}+z^{13})$.
And there is a similar question: Let $x$,$y$,$z$ be positive real numbers which satisfy the inequality
$(2x^4+3y^4)(2y^4+3z^4)(2z^4+3x^4) \leq(3x+2y)(3y+2z)(3z+2x)$. Prove this inequality:
$xyz\leq 1$.
 [/INST] These inequalities are algebraic and thus can be proved purely algorithmically.
Mathematica takes a minute or two for this proof of your first inequality:


Here is a "more human" proof:
Substituting $z=\frac1{xy}$, rewrite your first inequality as
\begin{equation}
    f(x,y)\mathrel{:=}\left(\frac{1}{x^{10} y^{10}}+x^{10}+y^{10}\right)^2-3 \left(\frac{1}{x^{13} y^{13}}+x^{13}+y^{13}\right)\ge0
\end{equation}
and then as
\begin{align}
g(x,y)&\mathrel{:=}f(x,y)x^{20} y^{20} \\ 
&=x^{40} y^{20}-3 x^{33} y^{20}+2 x^{30} y^{30}+x^{20} y^{40}-3 x^{20} y^{33} \\ 
&+2 x^{20} y^{10}+2 x^{10}
   y^{20}-3 x^7 y^7+1\ge0,
\end{align}
for $x,y>0$.
Further,
\begin{align}
g_1(x,y)\mathrel{:=}{}\frac{g'_x(x,y)}{x^6 y^7}&=40 x^{33} y^{13}-99 x^{26} y^{13}+60 x^{23} y^{23}+20 x^{13} y^{33} \\ 
&-60 x^{13} y^{26}+40 x^{13} y^3+20 x^3
   y^{13}-21, \\
g_2(x,y)\mathrel{:=}\frac{g'_y(x,y)}{x^7 y^6}&=20 x^{33} y^{13}-60 x^{26} y^{13}+60 x^{23} y^{23}+40 x^{13} y^{33} \\ 
&-99 x^{13} y^{26}+20 x^{13} y^3+40 x^3
   y^{13}-21,
\end{align}
and the only positive roots of the resultants of $g_1(x,y)$ and $g_2(x,y)$ with respect to $x$ and $y$ are $y=1$ and $x=1$, respectively. So, $(1,1)$ is the only critical point of $g$.
Next, all the coefficients of the polynomial $g(1+u,1+v)$ in $u,v$ are nonnegative. Therefore and because of the symmetry $x\leftrightarrow y$, it remains to consider the cases (i) $0\le x\le1$ and $y>0$ is large enough and (ii) $x=0$.
For case (i), we have $g(x,y)\ge1 - 3 x^7 y^7 + 2 x^{10} y^{20}>0$. For case (ii), we have $g(0,y)=1>0$.
So, your first inequality is proved, again.

It took Mathematica about 1.8 hours to prove your second inequality (click on the image to enlarge it):

The latter proof would probably take many thousands pages.
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