Is $a,b,c$ are three different positive integers such that :
$$ ab+bc+ca\geq 107 $$
Then what is the minimum value of $a^3+b^3+c^3-3abc$.
I expanded this expression to $(a+b+c)((a+b+c)^2-3(ab+bc+ca))$ and tried to find the minimum value of $(a+b+c)$ by AM-GM inequalities but the problem is that the minimum value occurs at scenario where $a=b=c$, which is ommitted by assumptions of this question. I'd appreciate some help.