Question

[Anh Khương Vũ Phương]

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

[Lightning Farron]

Áp dụng BĐT Cauchy-Schwarz ta có:

\(\dfrac{a}{a+\sqrt{\left(a+b\right)\left(a+c\right)}}\le\dfrac{a}{a+\sqrt{\left(\sqrt{ac}+\sqrt{ab}\right)^2}}=\dfrac{\sqrt{a}}{\sqrt{a}+\sqrt{c}+\sqrt{b}}\)

Tương tự cho 2 BĐT còn lại rồi cộng theo vế:

\(VT\le\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{a}+\sqrt{b}+\sqrt{c}}=1=VP\)