Question

[Lâm Ngọc]

Cho a,b,c>0 thoa mãn abc=1.CMR\(\frac{a}{\left(a+1\right)\left(b+1\right)}+\frac{b}{\left(b+1\right)\left(c+1\right)}+\frac{c}{\left(c+1\right)\left(a+1\right)}\ge\frac{3}{4}\)3/4     

Answer 1

[Kiệt Nguyễn]

Ta có: \(\frac{a}{\left(a+1\right)\left(b+1\right)}+\frac{b}{\left(b+1\right)\left(c+1\right)}+\frac{c}{\left(c+1\right)\left(a+1\right)}\ge\frac{3}{4}\)\(\Leftrightarrow\frac{a\left(c+1\right)+b\left(a+1\right)+c\left(b+1\right)}{\left(a+1\right)\left(b+1\right)\left(c+1\right)}\ge\frac{3}{4}\)\(\Leftrightarrow4\left(ab+bc+ca\right)+4\left(a+b+c\right)\ge3abc+3\left(ab+bc+ca\right)+3\left(a+b+c\right)+3\)\(\Leftrightarrow ab+bc+ca+a+b+c\ge6\)(abc = 1)

Bất đẳng thức cuối đúng theo bất đẳng thức Cô - si nên ta có điều phải chứng minh

Đẳng thức xảy ra khi a = b = c = 1