Violympic toán 9
Các câu hỏi tương tự
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.\(\frac{1}{1+a^3}+\frac{1}{1+b^3}+\frac{1}{1+c^3}\ge\frac{3}{1+abc}\)
17 tháng 11 2019 lúc 17:52
1. \(\left\{{}\begin{matrix}x,y,z>0\\xyz=1\end{matrix}\right.\) Cmr: \(\frac{x^2}{\left(x+1\right)^2}+\frac{y^2}{\left(y+1\right)^2}+\frac{z^2}{\left(z+1\right)^2}\ge\frac{3}{4}\)\
2. \(a,b,c>0.\) cmr: \(\Sigma\frac{a^3}{\left(2a^2+b^2\right)\left(2a^2+c^2\right)}\le\frac{1}{a+b+c}\)
25 tháng 4 2020 lúc 12:03
2. a) left{{}begin{matrix}x,y,z1x+y+zxyzend{matrix}right. Tìm min Pfrac{x-1}{y^2}+frac{y-1}{z^2}+frac{z-1}{x^2}
b) a,b,c0.Cmr:left(frac{a}{b}+frac{b}{c}+frac{c}{a}right)^2geleft(a+b+cright)left(frac{1}{a}+frac{1}{b}+frac{1}{c}right)
c) left{{}begin{matrix}x,y,zge0x^2+y^2+z^22end{matrix}right. Tìm max Pfrac{x^2}{x^2+yz+x+1}+frac{y+z}{x+y+z+1}-frac{1+yz}{9}
d) left{{}begin{matrix}a,b,c0a+b+c3end{matrix}right.. Cmr: frac{a}{ab+3c}+frac{b}{bc+3a}+frac{c}{ca+3b}gefrac{3}{4}
Đọc tiếp
2. a) \(\left\{{}\begin{matrix}x,y,z>1\\x+y+z=xyz\end{matrix}\right.\) Tìm min \(P=\frac{x-1}{y^2}+\frac{y-1}{z^2}+\frac{z-1}{x^2}\)
b) \(a,b,c>0.Cmr:\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^2\ge\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
c) \(\left\{{}\begin{matrix}x,y,z\ge0\\x^2+y^2+z^2=2\end{matrix}\right.\) Tìm max \(P=\frac{x^2}{x^2+yz+x+1}+\frac{y+z}{x+y+z+1}-\frac{1+yz}{9}\)
d) \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=3\end{matrix}\right.\). Cmr: \(\frac{a}{ab+3c}+\frac{b}{bc+3a}+\frac{c}{ca+3b}\ge\frac{3}{4}\)
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
a) cho x,y dương. CMR: \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\)
b) cho a+b+c=1 CMR: \(\frac{a}{a+b^2}+\frac{b}{b+c^2}+\frac{c}{c+a^2}\le\frac{1}{4}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
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 : \(Q=\frac{a+1}{a^2+2a+2}+\frac{b+1}{b^2+2b+2}+\frac{c+1}{c^2+2c+2}\)
1/CMR
a/x^4-2x^3+2x^2-2x+1ge0forall xin R
b/cho age0,bge2,a+b+c3. CMR : a^2+b^2+c^2le5
c/cho a,b,c 0 . CMR : frac{b+c}{a}+frac{a+c}{b}+frac{a+b}{c}ge4left(frac{a}{b+c}+frac{b}{a+c}+frac{c}{a+b}right)
2/ cho x,yge0,x+y1. tìm GTLN,GTNN của A x^2+y^2
3/ cho x,y0 .tìm GTNN của B frac{left(x+yright)^2}{x^2+y^2}+frac{left(x+yright)^2}{xy}
Đọc tiếp
1/CMR
a/\(x^4-2x^3+2x^2-2x+1\ge0\forall x\in R\)
b/cho \(a\ge0,b\ge2,a+b+c=3\). CMR : \(a^2+b^2+c^2\le5\)
c/cho a,b,c >0 . CMR : \(\frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}\ge4\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right)\)
2/ cho \(x,y\ge0,x+y=1\). tìm GTLN,GTNN của A =\(x^2+y^2\)
3/ cho x,y>0 .tìm GTNN của B= \(\frac{\left(x+y\right)^2}{x^2+y^2}+\frac{\left(x+y\right)^2}{xy}\)
8 tháng 8 2019 lúc 10:16
1. a) a,b,c0. Cmr: Sigmafrac{19b^3-a^3}{ab+5b^2}le3left(a+b+cright)
b) left{{}begin{matrix}a,b0a^2+b^2ge6end{matrix}right. . Cmr: sqrt{3left(a^2+6right)}gesqrt{2}left(a+bright)
2. a) left{{}begin{matrix}x,y,z0x^2ge y^2+z^2end{matrix}right.. Tìm Min Ax^2left(frac{1}{y^2}+frac{1}{z^2}right)+frac{y^2+z^2}{x^2}
b) left{{}begin{matrix}0le a,b,cle2a+b+c3end{matrix}right.. Tìm Max Pa^2+b^2+c^2
Ai bt giúp mk vs ! Mk cần trước 3h chiều nay ,Cảm ơn!
Đọc tiếp
1. a) \(a,b,c>0\). Cmr: \(\Sigma\frac{19b^3-a^3}{ab+5b^2}\le3\left(a+b+c\right)\)
b) \(\left\{{}\begin{matrix}a,b>0\\a^2+b^2\ge6\end{matrix}\right.\) . Cmr: \(\sqrt{3\left(a^2+6\right)}\ge\sqrt{2}\left(a+b\right)\)
2. a) \(\left\{{}\begin{matrix}x,y,z>0\\x^2\ge y^2+z^2\end{matrix}\right.\). Tìm Min \(A=x^2\left(\frac{1}{y^2}+\frac{1}{z^2}\right)+\frac{y^2+z^2}{x^2}\)
b) \(\left\{{}\begin{matrix}0\le a,b,c\le2\\a+b+c=3\end{matrix}\right.\). Tìm Max \(P=a^2+b^2+c^2\)
Ai bt giúp mk vs ! Mk cần trước 3h chiều nay ,Cảm ơn!
a) CMR với mọi số thực x,y > 0 ta có \(x^3+y^3\ge xy\left(x+y\right)\)
b) Cho a,b,c là các số thực dương thỏa mãn điều kiện abc=1. CMR:
\(\frac{1}{a+b+4}+\frac{1}{b+c+4}+\frac{1}{c+a+4}\le\frac{1}{2}\)