\(VT=\dfrac{a}{c}+\dfrac{b}{c}+\dfrac{a}{b}+\dfrac{c}{b}+\dfrac{b}{a}+\dfrac{c}{a}\)
\(\dfrac{a}{c}+\dfrac{c}{a}\ge2\sqrt{\dfrac{a}{c}.\dfrac{c}{a}}=2\)
Làm tương tự với các nhóm còn lại rồi cộng với nhau, ta được đpcm
1)cho a,b,c >0. \(cmr:\dfrac{1}{a^2+bc}+\dfrac{1}{b^2+ca}+\dfrac{1}{c^2+ab}\le\dfrac{a+b+c}{2abc}\)
2) cho a,b,c>0 và a+b+c=1. \(cmr:\left(1+\dfrac{1}{a}\right)\left(1+\dfrac{1}{b}\right)\left(1+\dfrac{1}{c}\right)\ge64\)
3) cho a,b,c>0. \(cme:\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)
4) cho a,b,c>0 .\(cmr:\dfrac{a^3}{b^3}+\dfrac{b^3}{c^3}+\dfrac{c^3}{a^3}\ge\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\)
5)cho a,b,c>0. cmr: \(\dfrac{1}{a\left(a+b\right)}+\dfrac{1}{b\left(b+c\right)}+\dfrac{1}{c\left(c+a\right)}\ge\dfrac{27}{2\left(a+b+c\right)^2}\)
Cho a, b, c > 0. Chứng minh \(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)
Chứng minh các BĐT sau:
a. \(9\left(\dfrac{1}{a+2b}+\dfrac{2}{b+2c}+\dfrac{3}{c+2a}\right)\le\dfrac{7}{a}+\dfrac{4}{b}+\dfrac{7}{c}\)
b. \(\dfrac{1}{a}+\dfrac{2}{b}+\dfrac{3}{c}\ge\dfrac{3}{a+b}+\dfrac{18}{3b+4c}+\dfrac{9}{c+6a}\)
c. \(\dfrac{b+c}{a}+\dfrac{2a+c}{b}+\dfrac{4\left(a+b\right)}{a+c}\ge9\)
Chứng minh rằng với ba số dương a, b, c ta luôn có:\(\dfrac{a}{a\:+\:b}\:+\dfrac{b}{b\:+\:c}\:+\:\dfrac{c}{c\:+\:a}\:< \:\sqrt{\dfrac{c}{a\:+\:b}\:}\:+\:\sqrt{\dfrac{b}{c\:+\:a}}\:+\:\sqrt{\dfrac{a}{b\:+\:c}}\)
Giả sử a, b, c là độ dài ba cạnh của một tam giác. Chứng minh:
a) \(\dfrac{a}{b+c-a}+\dfrac{b}{c+a-b}+\dfrac{c}{a+b-c}\ge3\)
b) \(\dfrac{a}{a+b-c}+\dfrac{b}{b+c-a}+\dfrac{c}{c+a-b}\ge3\)
Chao a, b, c >0
CMR \(\left(a^3+b^3+c^3\right)\left(\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}\right)\ge\dfrac{3}{2}\left(\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}\right)\)
cho a,b,c là các số thực dương thỏa a+b+c=1. Cmr
\(\dfrac{1+a}{1-a}+\dfrac{1+b}{1-b}+\dfrac{1+c}{1-c}\le2\left(\dfrac{b}{a}+\dfrac{a}{c}+\dfrac{c}{b}\right)\)
Chứng minh các bất đẳng thức :
a / \(\dfrac{a}{b}+\dfrac{b}{a}>=2;\forall a,b>0\)
b / \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}>=3;\forall a,b,c>0\)
c / \(\left(a+b\right)\left(b+c\right)+\left(c+a\right)>=8abc;\forall a,b,c>=0\)
d / \(\left(a+b+c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)>=9,\forall a,b,c>0\)
e / \(\left(1+\dfrac{a}{b}\right)\left(1+\dfrac{b}{c}\right)+\left(1+\dfrac{c}{a}\right)>=8,\forall a,b,c>0\)
f / \(\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)>=4,\forall a,b,>0\)
HELP ME !!!!!!
Cho a,b,c>0.Chứng minh rằng:
\(\dfrac{a+b}{a^2+b^2}+\dfrac{b+c}{b^2+c^2}+\dfrac{a+c}{a^2+c^2}\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
Help me?!
