Question

Cho a,b,c là các số dương

CMR : \(\dfrac{a+b}{c}+\dfrac{b+c}{a}+\dfrac{a+c}{b}\ge4\left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)\)

Answer 1

Lời giải:

Áp dụng BĐT Cauchy-Schwarz:

\(\frac{1}{b}+\frac{1}{c}\geq \frac{4}{b+c}\)

\(\Rightarrow \frac{a}{b}+\frac{a}{c}\geq \frac{4a}{b+c}(1)\)

Hoàn toàn tương tự: \(\frac{b}{c}+\frac{b}{a}\geq \frac{4b}{c+a}(2)\)

\(\frac{c}{a}+\frac{c}{b}\geq \frac{4c}{a+b}(3)\)

Lấy \((1)+(2)+(3)\Rightarrow \frac{a}{b}+\frac{a}{c}+\frac{b}{c}+\frac{b}{a}+\frac{c}{a}+\frac{c}{b}\geq 4\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)\)

\(\Leftrightarrow \frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\geq 4\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)\)

Ta có đpcm

Dấu bằng xảy ra khi $a=b=c$