Cho a > b \(\ge0\)
CMR: a + \(\dfrac{1}{\left(b+1\right)^2\left(a-b\right)}\ge3\)
(Sử dụng Cauchy)
Cho a>b\(\ge\)0.CMR:
a+\(\dfrac{1}{\left(b+1\right)^2\left(a-b\right)}\ge3\)
(Sử dụng Cauchy)
Hứa tặng GP nha :))
I. BĐT:
1.Cho a,b,c là độ dài của ba cạnh tam giác CMR:
\(\left(a\right)a^2+b^2+c^2< 2\left(ab+bc+ca\right)\)
\(\left(b\right)\dfrac{a}{b+c-a}+\dfrac{b}{a+c-b}+\dfrac{c}{a+b-c}\ge3\)
\(\left(c\right)\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}< 2\)
2. Cho a, b, c, d > 0 và abcd = 1 CMR: \(a^2+b^2+c^2+d^2+ab+cd\ge6\)
3. \(\left(x-1\right)\left(x-3\right)\left(x-4\right)\left(x-6\right)+9\ge0\)
4. \(\dfrac{ab}{a+b}+\dfrac{bc}{b+c}+\dfrac{ca}{c+a}\le\dfrac{a+b +c}{2}\)
cho a,b,c là các số thực dương
Cmr: \(\dfrac{2a}{b}+\dfrac{2b}{c}+\dfrac{2c}{a}\ge3+\dfrac{\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a+b+c\right)^2}\)
cho a,b,c là các số thực . Cmr:
\(\dfrac{2a}{b+c}+\dfrac{2b}{a+c}+\dfrac{2c}{a+b}\ge3+\dfrac{\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a+b+c\right)^2}\)
Chứng minh: BĐT: \(\left(a+b+c\right)^2\ge3.\left(ab+bc+ca\right)\)
Cho \(\left(a^2-bc\right)\left(b-abc\right)=\left(b^2-ac\right)\left(a-abc\right);abc\ne0;a\ne b\)
CMR:\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=a+b+c\)
CMR nếu \(\left(a^2-bc\right).\left(b-abc\right)=\left(b^2-ac\right).\left(a-abc\right)\) và các số a, b, c, a-b khác 0 thì \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=a+b+c\)
Cho a,b, c>0 thỏa mãn a+b+c=3.
CMR: \(\dfrac{a^3}{\left(a+1\right)\left(b+1\right)}+\dfrac{b^3}{\left(b+1\right)\left(c+1\right)}+\dfrac{c^3}{\left(c+1\right)\left(a+1\right)}>=\dfrac{3}{4}\)
