Các câu hỏi tương tự
cho a,b,c>0 cm
\(\frac{2a}{b+c}+\frac{2b}{c+a}+\frac{2c}{a+b}\ge3+\frac{\left(a-b\right)^2+\left(b-c\right)^2\left(c-a\right)^2}{\left(a+b+c\right)^2}\)
Cho a,b,c là các số thực dương và abc = 1
CMR: \(\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2\ge3\left(a+b+c+1\right)\)
a, b, c > 0. CMR: \(\left(\frac{ab}{c}\right)^2+\left(\frac{bc}{a}\right)^2+\left(\frac{ac}{b}\right)^2\ge3\left(\frac{ab+bc+ac}{a+b+c}\right)^2\)
CMR: \(\left(2+\frac{a}{b}\right)^{\alpha}+\left(2+\frac{b}{c}\right)^{\alpha}+\left(2+\frac{c}{a}\right)^{\alpha}\ge3^{\alpha+1}\left(\forall a,b,c>0\right)\)
Cho a,b,c > 0. CMR:
\(\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge3\sqrt[3]{\frac{3\left(a+b\right)\left(b+c\right)\left(c+a\right)\left(a+b+c\right)}{\left(ab+bc+ca\right)^2}}\)
Cho các số dương a, b, c thỏa mãn ab+bc+ca=1.
CMR: \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge3+\sqrt{\frac{\left(a+b\right)\left(a+c\right)}{a^2}}+\sqrt{\frac{\left(b+c\right)\left(b+a\right)}{b^2}}+\sqrt{\frac{\left(c+a\right)\left(c+b\right)}{c^2}}\)
Với a,b,c >0 và không hai số nào bằng nhau. Chứng minh rằng:
\(\frac{a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)}{a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)}\ge3\sqrt[3]{abc}\)
cho a,b,c >0
chứng minh \(\left(1+\frac{1}{a}\right)^4+\left(1+\frac{1}{b}\right)^4+\left(1+\frac{1}{c}\right)^4\ge3.\left(1+\frac{3}{2+abc}\right)^4\)
Cho 3 số x,y,z>0 thỏa mãn:
\(\frac{4a^2+\left(b-c\right)^2}{2a^2+b^2+c^2}+\frac{4b^2+\left(c-a\right)^2}{2b^2+c^2+a^2}+\frac{4c^2+\left(a-b\right)^2}{2c^2+a^2+b^2}\ge3\)