\(a+b+c=abc\Rightarrow\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=1\)
\(A=\frac{a+b-2}{a^2}+\frac{b+c-2}{b^2}+\frac{c+a-2}{c^2}-\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
\(A=\frac{a-1}{a^2}+\frac{a-1}{c^2}+\frac{b-1}{a^2}+\frac{b-1}{b^2}+\frac{c-1}{b^2}+\frac{c-1}{c^2}\)
\(A=\left(a-1\right)\left(\frac{1}{a^2}+\frac{1}{c^2}\right)+\left(b-1\right)\left(\frac{1}{a^2}+\frac{1}{b^2}\right)+\left(c-1\right)\left(\frac{1}{b^2}+\frac{1}{c^2}\right)-\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
\(A\ge\frac{2\left(a-1\right)}{ac}+\frac{2\left(b-1\right)}{ab}+\frac{2\left(c-1\right)}{bc}-\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
\(A\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-2\)
\(A\ge\sqrt{3\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)}-2=\sqrt{3}-2\)
\(A_{min}=\sqrt{3}-2\) khi \(a=b=c=\sqrt{3}\)
