Các câu hỏi tương tự
Cho các số dương a, b, c thỏa mãn ab+bc+ca=1.
CMR: \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge3+\sqrt{\frac{\left(a+b\right)\left(a+c\right)}{a^2}}+\sqrt{\frac{\left(b+c\right)\left(b+a\right)}{b^2}}+\sqrt{\frac{\left(c+a\right)\left(c+b\right)}{c^2}}\)
\(\frac{a^2\left(b+1\right)}{a+b+ab}+\frac{b^2\left(c+1\right)}{b+c+bc}+\frac{c^2\left(a+1\right)}{c+a+ca}\)cho a,b,c > 0 và a+b+c =3 CMR :
Afrac{a^2+bc}{b+ac}+frac{b^2+ca}{c+ab}+frac{c^2+ab}{a+bc}frac{3left(a^2+bcright)}{left(a+b+cright)b+3ac}+frac{3left(b^2+caright)}{left(a+b+cright)c+3ab}+frac{3left(c^2+abright)}{left(a+b+cright)a+3bc}gefrac{3left(a^2+bcright)}{left(a^2+bcright)+left(b^2+caright)+left(c^2+abright)}+frac{3left(b^2+caright)}{left(a^2+bcright)+left(b^2+caright)+left(c^2+abright)}+frac{3left(c^2+abright)}{left(a^2+bcright)+left(b^2+caright)+left(c^2+abright)}3
Đọc tiếp
\(A=\frac{a^2+bc}{b+ac}+\frac{b^2+ca}{c+ab}+\frac{c^2+ab}{a+bc}\)
\(=\frac{3\left(a^2+bc\right)}{\left(a+b+c\right)b+3ac}+\frac{3\left(b^2+ca\right)}{\left(a+b+c\right)c+3ab}+\frac{3\left(c^2+ab\right)}{\left(a+b+c\right)a+3bc}\)
\(\ge\frac{3\left(a^2+bc\right)}{\left(a^2+bc\right)+\left(b^2+ca\right)+\left(c^2+ab\right)}+\frac{3\left(b^2+ca\right)}{\left(a^2+bc\right)+\left(b^2+ca\right)+\left(c^2+ab\right)}+\frac{3\left(c^2+ab\right)}{\left(a^2+bc\right)+\left(b^2+ca\right)+\left(c^2+ab\right)}=3\)
cho a,b,c là số dương : CMR
\(\frac{bc}{a^2\left(b+c\right)}+\frac{ca}{b^2\left(c+a\right)}+\frac{ab}{c^2\left(a+b\right)}\ge\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
Cho \(\frac{1}{a^2-bc}+\frac{1}{b^2-ca}+\frac{1}{c^2-ab}=0\)
CM:\(\frac{a}{\left(a^2-bc\right)^2}+\frac{b}{\left(b^2-ca\right)^2}+\frac{c}{\left(c^2-ab\right)}=0\)
Cho a,b,c>0 và abc=1
CMR \(\frac{a}{\left(ab+a+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ca+c+1\right)^2}\ge\frac{1}{a+b+c}\)
Cho abc=a+b+c ; a,b,c>0
Tính \(A=\frac{1}{ab}\sqrt{\frac{\left(a^2+1\right)\left(b^2+1\right)}{c^2+1}}+\frac{1}{bc}\sqrt{\frac{\left(b^2+1\right)\left(c^2+1\right)}{a^2+1}}+\frac{1}{ca}\sqrt{\frac{\left(c^2+1\right)\left(a^2+1\right)}{b^2+1}}\)
đặt Pfrac{1}{1-ab}+frac{1}{1-bc}+frac{1}{1-ca}Rightarrow P-3frac{ab}{1-ab}+frac{bc}{1-bc}+frac{ca}{1-ca}lefrac{ab}{1-frac{a^2+b^2}{2}}+frac{bc}{1-frac{b^2+c^2}{2}}+frac{ca}{1-frac{c^2+a^2}{2}}lefrac{1}{2}.frac{left(a+bright)^2}{left(a^2+c^2right)+left(b^2+c^2right)}+frac{1}{2}.frac{left(b+cright)^2}{left(a^2+b^2right)+left(c^2+a^2right)}+frac{1}{2}.frac{left(c+aright)^2}{left(b^2+c^2right)+left(b^2+a^2right)}lefrac{1}{2}.left(frac{a^2}{a^2+c^2}+frac{b^2}{b^2+c^2}+frac{b^2}{a^2+b^2}+frac{c^2}{c^2...
Đọc tiếp
đặt \(P=\frac{1}{1-ab}+\frac{1}{1-bc}+\frac{1}{1-ca}\)
\(\Rightarrow P-3=\frac{ab}{1-ab}+\frac{bc}{1-bc}+\frac{ca}{1-ca}\le\frac{ab}{1-\frac{a^2+b^2}{2}}+\frac{bc}{1-\frac{b^2+c^2}{2}}+\frac{ca}{1-\frac{c^2+a^2}{2}}\)
\(\le\frac{1}{2}.\frac{\left(a+b\right)^2}{\left(a^2+c^2\right)+\left(b^2+c^2\right)}+\frac{1}{2}.\frac{\left(b+c\right)^2}{\left(a^2+b^2\right)+\left(c^2+a^2\right)}+\frac{1}{2}.\frac{\left(c+a\right)^2}{\left(b^2+c^2\right)+\left(b^2+a^2\right)}\)
\(\le\frac{1}{2}.\left(\frac{a^2}{a^2+c^2}+\frac{b^2}{b^2+c^2}+\frac{b^2}{a^2+b^2}+\frac{c^2}{c^2+a^2}+\frac{c^2}{b^2+c^2}+\frac{a^2}{b^2+a^2}\right)=\frac{3}{2}\)
\(\Rightarrow P-3\le\frac{3}{2}\Rightarrow P\le\frac{9}{2}\)
Bài 1 :Cho a,b,c dương thỏa mãn a+b+c=2
CMR \(\frac{bc}{\sqrt{3a^2+4}}+\frac{ca}{\sqrt{3b^2+4}}+\frac{ab}{\sqrt{3c^2+4}}\ge\frac{\sqrt{3}}{3}\)
Bài 2:Cho a,b,c>0. CMR
\(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge\frac{8}{9}\left(a+b+c\right)\left(ab+bc+ca\right)\)