Question

Chứng minh : \(\dfrac{a+b+c}{\sqrt{a\left(a+3b\right)}+\sqrt{b\left(b+3c\right)}\sqrt{c\left(c+3a\right)}}\) ≥ \(\dfrac{1}{2}\) với a,b,c là các số dương  

Answer 1

\(\dfrac{a+b+c}{\sqrt{a\left(a+3b\right)}+\sqrt{b\left(b+3c\right)}+\sqrt{c\left(c+3a\right)}}\)

\(=\dfrac{2\left(a+b+c\right)}{\sqrt{4a\left(a+3b\right)}+\sqrt{4b\left(b+3c\right)}+\sqrt{4c\left(c+3a\right)}}\)

\(\ge\dfrac{2\left(a+b+c\right)}{\dfrac{4a+a+3b}{2}+\dfrac{4b+b+3c}{2}+\dfrac{4c+c+3a}{2}}\)

\(=\dfrac{2\left(a+b+c\right)}{4\left(a+b+c\right)}=\dfrac{1}{2}\left(đpcm\right)\)

- Dấu "=" xảy ra khi \(a=b=c\)