1) Cho a,b,c>0.Chứng minh $\frac{a^{4}+b^{4}+c^{4}}{ab+bc+ca}+\frac{3abc}{a+b+c}\geq \frac{2}{3}(a^{2}+b^{2}+c^{2})$
2)Cho a,b,c>0 thỏa mãn abc=1.Chứng minh $\frac{1}{a^{2}+ab+1}+\frac{1}{b^{2}+bc+1}+\frac{1}{c^{2}+ca+1}\leq \frac{1}{2}$
3)Cho a,b,c>,abc=1.Chứng minh $\frac{2}{a+b+c}+\frac{1}{3}\geq \frac{3}{ab+bc+ca}$
4)Cho a,b,c>0.Chứng minh $\frac{a^{2}}{b}+\frac{b^{2}}{a}+\frac{c^{2}}{a}\geq a+b+c+\frac{4(a-b)^{2}}{a+b+c}$
5)Cho 0≤a,b,c≤1.Chứng minh $ \frac{1}{1+a+c}+\frac{1}{1+b+c}+\frac{1}{1+c+a}\leq \frac{3}{1+2\sqrt[3]{abc}}$
6)Cho a,b,c>0.Chứng minh $\frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}\geq \frac{\sqrt{2}}{4}(\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}})$