Theo BĐT Bunhicopxki:
\(VT\ge\frac{\left(a^2+b^2+c^2\right)^2}{3\left(a^2+b^2+c^2\right)}=\frac{a^2+b^2+c^2}{3}=VP^{\left(Đpcm\right)}\)
Đúng 0
Bình luận (0)
Violympic toán 9
Các câu hỏi tương tự
Cho a,b,c dương và abc=1
CMR: \(\frac{a^4}{2\left(b+c\right)^2}+\frac{b^4}{2\left(a+c\right)^2}+\frac{c^4}{2\left(a+b\right)^2}+\frac{1}{c^2\left(a+c\right)\left(a+b\right)}+\frac{1}{b^2\left(a+b\right)\left(b+c\right)}+\frac{1}{a^2\left(a+c\right)\left(a+b\right)}\ge\frac{1}{8}\)
Cho a,b,c 0. CMR:
1. a^3+b^3+c^3ge3abc
2. frac{x^2}{a}+frac{y^2}{b}gefrac{left(x+yright)^2}{a+b}
3. frac{x^2}{a}+frac{y^2}{b}+frac{z^2}{c}gefrac{left(x+y+zright)^2}{a+b+c}
4. a^4+b^4+c^4ge abcleft(a+b+cright)
5. frac{1}{left(a+1right)^2}+frac{1}{left(b+1right)^2}gefrac{1}{ab+1}
6.frac{1}{1+a^3}+frac{1}{1+b^3}+frac{1}{1+c^3}gefrac{3}{1+abc}
Đọc tiếp
Cho a,b,c > 0. CMR:
1. \(a^3+b^3+c^3\ge3abc\)
2. \(\frac{x^2}{a}+\frac{y^2}{b}\ge\frac{\left(x+y\right)^2}{a+b}\)
3. \(\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}\ge\frac{\left(x+y+z\right)^2}{a+b+c}\)
4. \(a^4+b^4+c^4\ge abc\left(a+b+c\right)\)
5. \(\frac{1}{\left(a+1\right)^2}+\frac{1}{\left(b+1\right)^2}\ge\frac{1}{ab+1}\)
6.\(\frac{1}{1+a^3}+\frac{1}{1+b^3}+\frac{1}{1+c^3}\ge\frac{3}{1+abc}\)
Cho các số thực dương a,b,c thỏa mãn c\(\ge\)a.Chứng minh rằng:
\(\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+4\left(\frac{c}{c+a}\right)^2\ge\frac{3}{2}\)
Chứng minh rằng:
\(\frac{a^2+b^2}{\left(a-b\right)^2}+\frac{b^2+c^2}{\left(b-c\right)^2}+\frac{c^2+a^2}{\left(c-a\right)^2}\ge\frac{5}{2}\)
Cho a,b,c là các số phân biệt . Chứng minh
\(\left(a^2+b^2+c^2\right)\left[\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}\right]\ge\frac{9}{2}\)
19 tháng 10 2019 lúc 17:49
1. cho 0 ale ble c . Cmr: frac{2a^2}{b^2+c^2}+frac{2b^2}{c^2+a^2}+frac{2c^2}{a^2+b^2}lefrac{a^2}{b}+frac{b^2}{c}+frac{c^2}{a}
2. cho a,b,cge0. cmr: a^2+b^2+c^2+3sqrt[3]{left(abcright)^2}ge2left(ab+bc+caright)
3. a,b,c0. Cmr: sqrt{left(a^2b+b^2c+c^2aright)left(ab^2+bc^2+ca^2right)}ge abc+sqrt[3]{left(a^3+abcright)left(b^3+abcright)left(c^3+abcright)}
4. a,b,c0. Tìm Min Pleft(frac{a}{a+b}right)^4+left(frac{b}{b+c}right)^4+left(frac{c}{c+a}right)^4
Đọc tiếp
1. cho \(0< a\le b\le c\) . Cmr: \(\frac{2a^2}{b^2+c^2}+\frac{2b^2}{c^2+a^2}+\frac{2c^2}{a^2+b^2}\le\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)
2. cho \(a,b,c\ge0\). cmr: \(a^2+b^2+c^2+3\sqrt[3]{\left(abc\right)^2}\ge2\left(ab+bc+ca\right)\)
3. \(a,b,c>0.\) Cmr: \(\sqrt{\left(a^2b+b^2c+c^2a\right)\left(ab^2+bc^2+ca^2\right)}\ge abc+\sqrt[3]{\left(a^3+abc\right)\left(b^3+abc\right)\left(c^3+abc\right)}\)
4. \(a,b,c>0\). Tìm Min \(P=\left(\frac{a}{a+b}\right)^4+\left(\frac{b}{b+c}\right)^4+\left(\frac{c}{c+a}\right)^4\)
11 tháng 2 2020 lúc 21:04
1. left{{}begin{matrix}a,b,c0a+b+c1end{matrix}right.. Cmr: frac{ab}{sqrt{left(1-cright)^2left(1+cright)}}+frac{bc}{sqrt{left(1-aright)^2left(1+aright)}}+frac{ca}{sqrt{left(1-bright)^3left(1+bright)}}lefrac{3sqrt{2}}{8}
2. left{{}begin{matrix}a,b,c0a+b+cle1end{matrix}right.. Cmr: frac{1}{a^2+b^2+c^2}+frac{1}{ableft(a+bright)}+frac{1}{bcleft(b+cright)}+frac{1}{acleft(a+cright)}gefrac{87}{2}
3. left{{}begin{matrix}a,b,c0ab+bc+ca2abcend{matrix}right.. Cmr: frac{1}{aleft(2a-1right)^2}+frac{1}{bleft...
Đọc tiếp
1. \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=1\end{matrix}\right.\). Cmr: \(\frac{ab}{\sqrt{\left(1-c\right)^2\left(1+c\right)}}+\frac{bc}{\sqrt{\left(1-a\right)^2\left(1+a\right)}}+\frac{ca}{\sqrt{\left(1-b\right)^3\left(1+b\right)}}\le\frac{3\sqrt{2}}{8}\)
2. \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c\le1\end{matrix}\right.\). Cmr: \(\frac{1}{a^2+b^2+c^2}+\frac{1}{ab\left(a+b\right)}+\frac{1}{bc\left(b+c\right)}+\frac{1}{ac\left(a+c\right)}\ge\frac{87}{2}\)
3. \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca=2abc\end{matrix}\right.\). Cmr: \(\frac{1}{a\left(2a-1\right)^2}+\frac{1}{b\left(2b-1\right)^2}+\frac{1}{c\left(2c-1\right)^2}\ge\frac{1}{2}\)
4. \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z=2015\end{matrix}\right.\). Tìm min \(A=\frac{x^4+y^4}{x^3+y^3}+\frac{y^4+z^4}{y^3+z^3}+\frac{z^4+x^4}{z^2+x^2}\)
Mn giúp mk với ạ! Thanks nhiều
28 tháng 10 2019 lúc 23:28
1. Cho a,b,c>0. Cmr: \(\frac{a}{\left(b+c\right)^2}+\frac{b}{\left(c+a\right)^2}+\frac{c}{\left(a+b\right)^2}\ge\frac{9}{4\left(a+b+c\right)}\)
Cho a,b,c > 0 , \(a^2+b^2+c^2=3\). Chứng minh rằng : \(\sqrt{\frac{9}{\left(a+b\right)^2}+c^2}+\sqrt{\frac{9}{\left(b+c\right)^2}+a^2}+\sqrt{\frac{9}{\left(a+c\right)^2}+b^2}\)≥\(\frac{3\sqrt{13}}{2}\)