\(\frac{a^2}{b+3c}+\frac{b^2}{c+3a}+\frac{c^2}{a+3b}\ge\frac{\left(a+b+c\right)^2}{4a+4b+4c}=\frac{a+b+c}{4}\)
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Các câu hỏi tương tự
Cho ba số thực dương a,b,c. CMR:
\(\frac{a^2}{b+3c}+\frac{b^2}{c+3a}+\frac{c^2}{a+3b}\ge\frac{a+b+c}{4}\)
Đẳng thức xảy ra khi nào?
Cho a,b,c>0. CMR: \(\frac{a^2}{\sqrt{3a^2+8b^2+14ab}}+\frac{b^2}{\sqrt{3b^2+8c^2+14bc}}+\frac{c^2}{\sqrt{3c^2+8a^2+14ca}}\ge\frac{a+b+c}{5}\)
Bai 1: Cho a,b,c \(\varepsilon\)R va a+b+c=1
CM: \(a^2+b^2+c^2\ge\frac{1}{3}\)
CM theo bdt BUNHIA COPXKI gium minh nha cac ban
\(\sqrt{\frac{bc}{a\left(3b+a\right)}}+\sqrt{\frac{ac}{b\left(3c+b\right)}}+\sqrt{\frac{ab}{c\left(3a+c\right)}}\ge\frac{3}{2}\)
a,b,c>0, biết a+b+c=3
CMR a)\(\frac{ab}{\sqrt{a^2+3b^2}}+\frac{bc}{\sqrt{b^2+3c^2}}+\frac{ac}{\sqrt{c^2+3a^2}}\)≤\(\frac{3}{2}\)
b)\(\frac{a}{\sqrt{b^2+3}}+\frac{b}{\sqrt{c^2+3}}+\frac{c}{\sqrt{a^2+3}}\)≥\(\frac{3}{2}\)
cho a,b,c>0 . CMR: \(\frac{b}{a+3b}+\frac{c}{b+3c}+\frac{a}{c+3a}\le\frac{a+b+c}{4}\)
Cho a, b, c dương. Chứng minh: \(\frac{1}{a\sqrt{3a+2b}}+\frac{1}{b\sqrt{3b+2c}}+\frac{1}{c\sqrt{3c+2a}}\ge\frac{3}{\sqrt{5abc}}\)
CM: \(\frac{\left(a^2+b^2+c^2\right)^2}{\left(a+b+c\right)^2}\ge\frac{a^2+b^2+c^2}{3}\)
Áp dung Bunhia nha các bạn
BL
18 tháng 11 2019 lúc 20:12
1.left{{}begin{matrix}a,b,c0ab+bc+ca3end{matrix}right. Cmr: frac{1}{a^2+1}+frac{1}{b^2+1}+frac{1}{c^2+1}gefrac{3}{2}
2.a,b,c0. Cmr: frac{ab^2}{a^2+2b^2+c^2}+frac{bc^2}{b^2+2c^2+a^2}+frac{ca^2}{c^2+2a^2+b^2}lefrac{a+b+c}{4}
3. a,b,c0. Cmr: frac{ab}{a+3b+2c}+frac{bc}{b+3c+2a}+frac{ca}{c+3a+2b}lefrac{a+b+c}{6}
Đọc tiếp
1.\(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca=3\end{matrix}\right.\) Cmr: \(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge\frac{3}{2}\)
2.\(a,b,c>0\). Cmr: \(\frac{ab^2}{a^2+2b^2+c^2}+\frac{bc^2}{b^2+2c^2+a^2}+\frac{ca^2}{c^2+2a^2+b^2}\le\frac{a+b+c}{4}\)
3. \(a,b,c>0\). Cmr: \(\frac{ab}{a+3b+2c}+\frac{bc}{b+3c+2a}+\frac{ca}{c+3a+2b}\le\frac{a+b+c}{6}\)