\(VT=\frac{a^2}{b^2}+\frac{b^2}{a^2}-4\left(\frac{a}{b}+\frac{b}{a}\right)+2+4+\left(\frac{a}{b}+\frac{b}{a}\right)-2\)
\(\Leftrightarrow VT=\left(2-\frac{a}{b}-\frac{b}{a}\right)^2+\left(\frac{a}{b}+\frac{b}{a}\right)-2\)
Theo Cosi có \(\frac{a}{b},\frac{b}{a}\) là hai số nghịch đảo nên \(\frac{a}{b}+\frac{b}{a}\ge2\Leftrightarrow\left(\frac{a}{b}+\frac{b}{a}\right)-2\ge0\)
Vậy VT >= 0 với a,b khác 0
cho: \(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0\) . CMR: \(\frac{a}{\left(b-c\right)^2}+\frac{b}{\left(c-a\right)^2}+\frac{c}{\left(a-b\right)^2}=0\)
Cho 3 số thực a,b,c thỏa mãn \(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}\) = 0. CMR
\(\frac{a}{\left(b-c\right)^2}+\frac{b}{\left(c-a\right)^2}+\frac{c}{\left(a-b\right)^2}\) = 0
cho a,b,c>0 và a+b+c=1. Cmr:
\(\frac{a^2}{a^2+\left(b+c\right)^2}+\frac{b^2}{b^2+\left(a+c\right)^2}+\frac{c^2}{c^2+\left(a+b\right)^2}\ge\frac{3}{5}\)
Cho a,b,c là 3 số thực thỏa mãn \(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}\). CMR : \(\frac{a}{\left(b-c\right)^2}+\frac{b}{\left(c-a\right)^2}+\frac{c}{\left(a-b\right)^2}=0\)
1. Cho a,b,c > 0. Cmr :
\(\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\ge\frac{3\left(a^2+b^2+c^2\right)}{a+b+c}\)
2. Cho a,b,c > 0. Cmr :
\(\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c}\ge\frac{2}{3}\)
Cho a, b, c là 3 số thực dương. CMR
\(\frac{\left(a+b\right)^2}{ab}+\frac{\left(b+c\right)^2}{bc}+\frac{\left(c+a\right)^2}{ca}\ge9+2\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)\)
Cho a,b,c > 0. CMR P = \(\frac{a^2}{b\left(b+2c\right)}+\frac{b^2}{c\left(c+2a\right)}+\frac{c^2}{a\left(a+2b\right)}\) ≥ 1
Cho các số thực dương a,b,c. CMR
\(\frac{\left(b+c-a\right)^2}{\left(b+c\right)^2+a^2}+\frac{\left(a+c-b\right)^2}{\left(a+c\right)^2+b^2}+\frac{\left(b+a-c\right)^2}{\left(b+a\right)^2+c^2}\ge\frac{3}{5}\)
Cho a,b,c > 0 và a + b + c = 1
CMR:
\(\left(a+\frac{1}{a}\right)^2+\left(b+\frac{1}{b}\right)^2+\left(c+\frac{1}{c}\right)^2>33\)
