<s>[INST] Logical suite and inequalities This is a related problem (see here)
We have the following inequalities :

For $n=3$ with $a,b,c$ real numbers the following inequality holds.
  $$\frac{1}{(a-b)^2}+\frac{1}{(c-b)^2}+\frac{1}{(a-c)^2}+(c-b)^2+(c-a)^2+(a-b)^2\geq \sqrt{54}$$

For $n=4$ with $a,b,c,d$ real numbers
$$\frac{1}{(a-d)^2}+\frac{1}{(d-b)^2}+\frac{1}{(d-c)^2}+\frac{1}{(a-b)^2}+\frac{1}{(c-b)^2}+\frac{1}{(a-c)^2}+(d-b)^2+(c-d)^2+(a-d)^2+(a-b)^2+(b-c)^2+(a-c)^2\geq \sqrt{288}$$
If we continue with $n=5,6,7\cdots$ there is a logical suite but I don't know how to prove this and what is the following numbers .
Thanks.
 [/INST] Your first inequality  it's just AM-GM.
Indeed, let $a=\min\{a,b,c\}$, $b=a+x$ and $c=a+y$.
Thus, $x>0$ and $y>0$ and 
$$\sum_{cyc}\left((a-b)^2+\frac{1}{(a-b)^2}\right)=x^2+y^2+(x-y)^2+\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{(x-y)^2}=$$
$$=2(x^2-xy+y^2)+\frac{(x^2-xy+y^2)^2}{x^2y^2(x-y)^2}\geq2\sqrt{\frac{2(x^2-xy+y^2)^3}{x^2y^2(x-y)^2}}=$$
$$=2\sqrt{\frac{2\left((x-y)^2+2\frac{xy}{2}\right)^3}{x^2y^2(x-y)^2}}\geq2\sqrt{\frac{2\left(3\sqrt[3]{(x-y)^2\cdot\left(\frac{xy}{2}\right)^2}\right)^3}{x^2y^2(x-y)^2}}=\sqrt{54}$$
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