Question

CMR a/b+c + b/a+c + c/a+b >= 3/2

Answer 1

\(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\ge\frac{3}{2}\)

Đặt b+c =x ; a+c =y ; a+b =z 

=> a =\(\frac{y+z-x}{2}\); b =\(\frac{x+z-y}{2}\);c =\(\frac{y+x-z}{2}\)

=> VT = \(\frac{1}{2}\left(\frac{y}{x}+\frac{z}{x}-1+\frac{x}{y}+\frac{z}{y}-1+\frac{y}{z}+\frac{x}{z}-1\right)=\frac{1}{2}\left(\left(\frac{y}{x}+\frac{x}{y}\right)+\left(\frac{z}{x}+\frac{x}{z}\right)+\left(\frac{z}{y}+\frac{y}{z}\right)-3\right)\)

VT \(\ge\frac{1}{2}\left(2+2+2-3\right)=\frac{3}{2}\left(đpcm\right)\)