Question

(a+b)/c + (b+c)/a + (c+a)/b mai mare sau egal cu 6
Aducand la numitor comun si eliminandu-l ajungem la a2b+ab2+b2c+bc2+ac2+a2c-6abc mai mare sau egal cu 0.
Ce fac in continuare ?

Answer 1

$\displaystyle Inegalitatea~initiala~este~echivalenta~cu \\ \\ \left( \frac{a}{b}+ \frac{b}{a} \right)+\left( \frac{b}{c}+ \frac{c}{b} \right)+\left( \frac{a}{c}+ \frac{c}{a} \right) \ge 6. \\ \\ Insa~se~cunoaste~inegalitatea~ \frac{x}{y}+ \frac{y}{x} \ge 2,~unde~x,y\ \textgreater \ 0. \\ \\ Conform~acestei~inegalitati,~fiecare~paranteza~este \ge 2,~si~deci \\ \\ inegalitatea~initiala~este~demonstrata.$