\( \displaystyle (2+\frac{1}{a})(2+\frac{1}{b})(2+\frac{1}{c}) \)
\( \displaystyle =(2+\frac{a+b+c}{a})(2+\frac{a+b+c}{b})(2+\frac{a+b+c}{c}) \)
\( \displaystyle =(3+\frac{b+c}{a})(3+\frac{c+a}{b})(3+\frac{a+b}{c}) \)
\( \displaystyle =3(\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab})+13(\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c})+38 \)
\( \displaystyle \ge 3 \cdot 3 \root 3 \of{\frac{a^2}{bc}\cdot \frac{b^2}{ca}\cdot \frac{c^2}{ab}}+13(2 \sqrt{\frac{a}{b} \cdot \frac{b}{a}}+2 \sqrt{\frac{b}{c} \cdot \frac{c}{b}}+2 \sqrt{\frac{c}{a} \cdot \frac{a}{c}})+38 \)
\( =9+78+38=125 \)
當\( \displaystyle a=b=c=\frac{1}{3} \)時等號成立
[ 本帖最後由 bugmens 於 2010-8-28 06:03 AM 編輯 ]
