Violympic toán 9
Các câu hỏi tương tự
a)chứng minh rằng: \(a^2-ab+b^2\ge\frac{1}{3}\left(a^2+ab+b^2\right)\) với mọi giá trị của a,b
b) cho các số dương a,b,c >0 cmr \(\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\ge\frac{a+b+c}{3}\)
Cho a , b , c > 0 thỏa mãn \(a^2b+b^2c+c^2a=3\)
Chứng minh \(\frac{ab+bc+ca}{2\left(a^2+b^2+c^2\right)}+\frac{1}{6}\left(\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}\right)\ge\frac{a+b+c}{3}\)
Cho a,b,c>0. Chứng minh rằng:
\(\frac{a^6}{b^3\left(c+a\right)}+\frac{b^6}{c^3\left(a+b\right)}+\frac{c^6}{a^3\left(b+c\right)}\ge\frac{ab+bc+ca}{2}\)
BL
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 \(P=\left(\frac{a}{a+b}\right)^4+\left(\frac{b}{b+c}\right)^4+\left(\frac{c}{c+a}\right)^4\)
BL
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
Cho a,b,c>0 và \(a^2b+b^2c+c^2a=3\)
Chứng minh rằng : \(\frac{ab+bc+ca}{2\left(a^2+b^2+c^2\right)}+\frac{1}{6}\left(\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab}\right)\)≥\(\frac{a+b+c}{3}\)
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: \(\frac{ab}{a+3b+2c}+\frac{bc}{b+3c+2a}+\frac{ca}{c+3a+2b}\le\frac{a+b+c}{6}\)
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.\(\frac{1}{1+a^3}+\frac{1}{1+b^3}+\frac{1}{1+c^3}\ge\frac{3}{1+abc}\)
BL
2 tháng 3 2020 lúc 23:45
1. a) left{{}begin{matrix}x,y,z0xyz1end{matrix}right.. Tìm max Pfrac{1}{sqrt{x^5-x^2+3xy+6}}+frac{1}{sqrt{y^5-y^2+3yz+6}}+frac{1}{sqrt{z^5-z^2+zx+6}}
b) left{{}begin{matrix}x,y,z0xyz8end{matrix}right.. Min Pfrac{x^2}{sqrt{left(1+x^3right)left(1+y^3right)}}+frac{y^2}{sqrt{left(1+y^3right)left(1+z^3right)}}+frac{z^2}{sqrt{left(1+z^3right)left(1+x^3right)}}
c) x,y,z0. Min Psqrt{frac{x^3}{x^3+left(y+zright)^3}}+sqrt{frac{y^3}{y^3+left(z+xright)^3}}+sqrt{frac{z^3}{z^3+left(x+yright)^3}}
d) a,b,c0;...
Đọc tiếp
1. a) \(\left\{{}\begin{matrix}x,y,z>0\\xyz=1\end{matrix}\right.\). Tìm max \(P=\frac{1}{\sqrt{x^5-x^2+3xy+6}}+\frac{1}{\sqrt{y^5-y^2+3yz+6}}+\frac{1}{\sqrt{z^5-z^2+zx+6}}\)
b) \(\left\{{}\begin{matrix}x,y,z>0\\xyz=8\end{matrix}\right.\). Min \(P=\frac{x^2}{\sqrt{\left(1+x^3\right)\left(1+y^3\right)}}+\frac{y^2}{\sqrt{\left(1+y^3\right)\left(1+z^3\right)}}+\frac{z^2}{\sqrt{\left(1+z^3\right)\left(1+x^3\right)}}\)
c) \(x,y,z>0.\) Min \(P=\sqrt{\frac{x^3}{x^3+\left(y+z\right)^3}}+\sqrt{\frac{y^3}{y^3+\left(z+x\right)^3}}+\sqrt{\frac{z^3}{z^3+\left(x+y\right)^3}}\)
d) \(a,b,c>0;a^2+b^2+c^2+abc=4.Cmr:2a+b+c\le\frac{9}{2}\)
e) \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=3\end{matrix}\right.\). Cmr: \(\frac{a}{b^3+ab}+\frac{b}{c^3+bc}+\frac{c}{a^3+ca}\ge\frac{3}{2}\)
f) \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca+abc=4\end{matrix}\right.\) Cmr: \(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\le3\)
g) \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca+abc=2\end{matrix}\right.\) Max : \(Q=\frac{a+1}{a^2+2a+2}+\frac{b+1}{b^2+2b+2}+\frac{c+1}{c^2+2c+2}\)