cho a,b,c>0

Cm: \(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2\sqrt[3]{abc}}\ge\frac{\left(a+b+c+\sqrt[3]{abc}\right)^2}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

cho a,b,c>0 và abc=1

Cmr: \(\frac{1}{\left(a+1\right)^2+b^2+1}+\frac{1}{\left(b+1\right)^2+c^2+1}+\frac{1}{\left(c+1\right)^2+a^2+1}\le\frac{1}{2}\)

Cho \(x\ge1,y\ge1\)

Cmr: \(\frac{1}{1+x^2}+\frac{1}{1+y^2}\ge\frac{2}{1+xy}\)

cho a,b,c>0 và \(a+b+c\le3\)

Cmr: \(\frac{1}{a^2+b^2+c^2}+\frac{2018}{ab+bc+ca}\ge672\)

cho x>1,y>0 .Cmr: \(\frac{1}{\left(x-1\right)^3}+\frac{\left(x-1\right)^3}{y^3}+\frac{1}{y^3}\ge3\left(\frac{3-2x}{x-1}+\frac{x}{y}\right)\)

cho các số thực không âm đôi một khác nhau thỏa mãn \(\left(x+z\right)\left(z+y\right)=1\)

Cmr: \(\frac{1}{\left(x-y\right)^2}+\frac{1}{\left(x+z\right)^2}+\frac{1}{\left(z+y\right)^2}\ge4\)

cho a,b,c>0. Cmr:

\(\sqrt{\frac{a^3}{a^3+\left(b+c\right)^3}}+\sqrt{\frac{b^3}{b^3+\left(a+c\right)^3}}+\sqrt{\frac{c^3}{c^3+\left(a+b\right)^3}}\ge1\)

cho \(x^2+4y^2=1\)

Cmr: \(\left|x-y\right|\le\frac{\sqrt{5}}{2}\)

cho a,b,c>0 và a+b+c=3

Tìm GTNN của \(A=\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\)

cho 0< a,b,c < 2 . Cmr: \(\frac{1}{2-a}+\frac{1}{2-b}+\frac{1}{2-c}\ge\frac{a^2+b^2+c^2}{2}+\frac{3}{2}\)