Áp dụng $AM-GM$ cho $3$ số và $2$ số ta được
$\begin{array}{l} 1 \ge a + b + c \ge 3\sqrt[3]{{abc}}\left( {AM - GM} \right)\\ \Rightarrow 3\sqrt[3]{{abc}} \le 1 \Rightarrow abc \le \dfrac{1}{{27}}\\ abc + \dfrac{1}{{abc}} = abc + \dfrac{1}{{729abc}} + \dfrac{{728}}{{729abc}} \ge 2\sqrt {\dfrac{{abc}}{{729abc}}} + \dfrac{{728}}{{729abc}}\\ \ge 2.\dfrac{1}{{27}} + \dfrac{{728}}{{729.\dfrac{1}{{27}}}} = \dfrac{{730}}{{27}}\\ \Rightarrow \min A = \dfrac{{730}}{{27}}\\ \Rightarrow abc = \dfrac{1}{{27}} \Rightarrow a = b = c = \dfrac{1}{3} \end{array}$
