<s>[INST] How to prove that $b^2c^2+c^2a^2+a^2b^2 \gt abc(a+b+c)$ I am stuck with the following problem.


If $a>0$, $b>0$, $c >0$ and not all equal then prove that: 
    $$b^2c^2+c^2a^2+a^2b^2 \gt abc(a+b+c).$$

Additional info:I'm looking for solutions  using  AM-GM . 

I don't know how to progress . 
I will be grateful if someone explains . Thanks in advance ..
 [/INST] From C-S:
$$\left(\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{a^2}\right)\left(a^2b^2+b^2c^2+c^2a^2\right)\ge \left(a+b+c\right)^2 \Rightarrow $$
$$\left(a^2b^2+b^2c^2+c^2a^2\right)^2\ge a^2b^2c^2(a+b+c)^2.$$
equality occurs when $a=b=c.$
Taking square root and $a\ne b\ne c$ will do.
</s>