cho a,b,c>0. Dat x= a+b+c. CMR:
$\dfrac{a}{b}$ + $\dfrac{b}{c}$ + $\dfrac{c}{a}$ $\dfrac{341x+a}{341x+b}$ +$\dfrac{341x+b}{341x+c}$+$\dfrac{341x+c}{341x+a}$
$\sum {\dfrac{a}{b}} \ge \sum {\dfrac{{341x + a}}{{341x + b}}} $
$ \Leftrightarrow \sum {\left( {\dfrac{a}{b} - \dfrac{{341x + a}}{{341x + b}}} \right)} \ge 0$
$\Leftrightarrow \sum {\dfrac{{341x(a - b)}}{{b(341x + b)}}} \ge 0$
$\sum {\dfrac{{a - b}}{{b(341x + b)}}} = (b - a)(b - c)\dfrac{{341x + b + c}}{{bc(341x + b)(341x + c)}} + {(a - c)^2}\dfrac{{341x + (a + c)}}{{ac(341x + a)(341x + c)}}$
Không mất tính tổng quát giả sử b là số nằm giữa suy ra ĐPCM