Question

Cho a, b, c dương. CMR:

\(\dfrac{a}{a+\sqrt{\left(a+b\right)\left(a+c\right)}}+\dfrac{b}{b+\sqrt{\left(b+c\right)\left(b+a\right)}}+\dfrac{c}{c+\sqrt{\left(c+a\right)\left(c+b\right)}}\le1\)

 

 

Answer 1

Áp dụng bđt Bu-nhi-a, ta có 

\(\sqrt{\left(a+b\right)\left(a+c\right)}\ge\sqrt{ab}+\sqrt{ac}\)

=>\(\frac{a}{a+\sqrt{\left(a+b\right)\left(a+c\right)}}\le\frac{a}{a+\sqrt{ab}+\sqrt{ac}}=\frac{\sqrt{a}}{\sqrt{a}+\sqrt{b}+\sqrt{c}}\)

Tương tự, rồi + vào, ta có 

A\(\le\frac{\sqrt{a}}{\sqrt{a}+\sqrt{b}+\sqrt{c}}+\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}+\sqrt{c}}+\frac{\sqrt{c}}{\sqrt{a}+\sqrt{b}+\sqrt{c}}=1\) (ĐPCM)

dấu =xảy ra <=>a=b=c>o

^_^