Chương 4: BẤT ĐẲNG THỨC, BẤT PHƯƠNG TRÌNH
Các câu hỏi tương tự
Cho a, b, c dương.
Cmr: \(\frac{1}{2a+b+c}+\frac{1}{a+2b+c}+\frac{1}{a+b+2c}\le\frac{1}{a+3b}+\frac{1}{b+3c}+\frac{1}{c+3a}\)
Cho a,b,c>0. CMR
\(\dfrac{1}{a+2b+c}+\dfrac{1}{b+2c+a}+\dfrac{1}{c+2a+b}\le\dfrac{1}{a+3b}+\dfrac{1}{b+3c}+\dfrac{1}{c+3a}\)
cho a,b,c > 0 thỏa mãn a+b+c=6abc.
Cmr: \(\frac{bc}{a^3\left(c+2b\right)}+\frac{ca}{b^3\left(a+2c\right)}+\frac{ab}{c^3\left(b+2a\right)}\ge2\)
cho a,b,c > 0 thỏa mãn \(a^2+b^2+c^2=1\) . Cmr:
\(\sqrt{\frac{ab+2c^2}{1+ab-c^2}}+\sqrt{\frac{bc+2a^2}{1+bc-a^2}}+\sqrt{\frac{ca+2b^2}{1+ac-b^2}}\ge2+ab+bc+ca\)
cho a,b,c là các số thực dương thỏa mãn ab+bc+ca=3. CMR:
\(\frac{a}{2a^2+bc}+\frac{b}{2b^2+ca}+\frac{c}{2c^2+ab}\ge abc\)
cho a,b,c> 0 . Cmr:
\(2\left(\frac{a}{b+2c}+\frac{b}{c+2a}+\frac{c}{a+2b}\right)\ge1+\frac{b}{b+2a}+\frac{c}{c+2b}+\frac{a}{a+2c}\)
cho \(c\ge b\ge a>0\) . Cmr: \(\frac{2a^2}{b+c}+\frac{2b^2}{c+a}+\frac{2c^2}{a+b}\le\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)
cho a,b,c là số thực dương thỏa mãn ab+bc+ac=abc
CMR: \(\frac{\sqrt{b^2+2a^2}}{ab}+\frac{\sqrt{c^2+2b^2}}{bc}+\frac{\sqrt{a^2+2c^2}}{ca}>\sqrt{3}\)
2. Cho a, b > 0. CM: \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)
Áp dụng CM các bđt sau:
a)Cho a, b, c > 0 thỏa mãn \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=4.\) CM:\(\frac{1}{2a+b+c}+\frac{1}{a+2b+c}+\frac{1}{a+b+2c}\le1\)
b)\(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\le\frac{a+b=c}{2}\left(a,b,c>0\right)\)