Violympic toán 9
Các câu hỏi tương tự
BL
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: \(\Sigma\frac{a^3}{\left(2a^2+b^2\right)\left(2a^2+c^2\right)}\le\frac{1}{a+b+c}\)
Cho 3 số thực dương x, y, z thỏa mãn \(x+y+z\le\frac{3}{2}\). Tìm GTNN của biểu thức:
\(P=\frac{x\left(yz+1\right)^2}{z^2_{ }\left(zx+1\right)}+\frac{y\left(zx+1\right)^2}{x^2\left(xy+1\right)}+\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}\)
Cho x,y,z>0 và x+y≤z
CMR: \(\left(x^2+y^2+z^2\right)\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)\ge\frac{27}{2}\)
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...
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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
BL
8 tháng 12 2019 lúc 16:44
Giải hpt : a) left{{}begin{matrix}left(x^2+y^2right)left(x+y+1right)25left(y+1right)x^2+xy+2y^2+x-8y9end{matrix}right. b) left{{}begin{matrix}x^2+y^2+6xy-frac{1}{left(x-yright)^2}+frac{9}{8}02y-frac{1}{x-y}+frac{5}{4}0end{matrix}right.
c) left{{}begin{matrix}frac{x}{x^2-y}+frac{5y}{x+y^2}45x+y+frac{x^2-5y^2}{xy}5end{matrix}right. d) left{{}begin{matrix}3xy+y+121x9x^2y^2+3xy+1117x^2end{matrix}right.
e) left{{}begin{matrix}xleft(x^2-y^2right)+x^21sqrt{left(x-y^2right)^3}7...
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Giải hpt : a) \(\left\{{}\begin{matrix}\left(x^2+y^2\right)\left(x+y+1\right)=25\left(y+1\right)\\x^2+xy+2y^2+x-8y=9\end{matrix}\right.\) b) \(\left\{{}\begin{matrix}x^2+y^2+6xy-\frac{1}{\left(x-y\right)^2}+\frac{9}{8}=0\\2y-\frac{1}{x-y}+\frac{5}{4}=0\end{matrix}\right.\)
c) \(\left\{{}\begin{matrix}\frac{x}{x^2-y}+\frac{5y}{x+y^2}=4\\5x+y+\frac{x^2-5y^2}{xy}=5\end{matrix}\right.\) d) \(\left\{{}\begin{matrix}3xy+y+1=21x\\9x^2y^2+3xy+1=117x^2\end{matrix}\right.\)
e) \(\left\{{}\begin{matrix}x\left(x^2-y^2\right)+x^2=1\sqrt{\left(x-y^2\right)^3}\\76x^2-20y^2+2=\sqrt[3]{4x\left(8x+1\right)}\end{matrix}\right.\)
tìm x, y > 0 sao cho \(\left(x^2+y+\frac{3}{4}\right)\left(y^2+x+\frac{3}{4}\right)=\left(2x+\frac{1}{2}\right)\left(2y+\frac{1}{2}\right)\)
Cho x, y >0 thỏa mãn : \(\frac{1}{3}< x\le\frac{1}{2};y\ge1.\) Tìm min A= \(x^2+y^2+\frac{x^2y^2}{\left(\left(4x-1\right)y-x\right)^2}\)
giải hệ phương trình:
1, left{{}begin{matrix}2yleft(4y^2+3x^2right)x^4left(x^2+3right)2012^xleft(sqrt{2y-2x+5}-x+1right)4024end{matrix}right.
2, left{{}begin{matrix}x^3-2x^2y-15x6yleft(2x-5-4yright)frac{x^2}{8y}+frac{2x}{3}sqrt{frac{x^3}{3y}+frac{x^2}{4}}-frac{y}{2}end{matrix}right.
3, left{{}begin{matrix}8left(x^2+y^2right)+4xy+frac{5}{left(x+yright)^2}132x+frac{1}{x+y}1end{matrix}right.
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giải hệ phương trình:
1, \(\left\{{}\begin{matrix}2y\left(4y^2+3x^2\right)=x^4\left(x^2+3\right)\\2012^x\left(\sqrt{2y-2x+5}-x+1\right)=4024\end{matrix}\right.\)
2, \(\left\{{}\begin{matrix}x^3-2x^2y-15x=6y\left(2x-5-4y\right)\\\frac{x^2}{8y}+\frac{2x}{3}=\sqrt{\frac{x^3}{3y}+\frac{x^2}{4}}-\frac{y}{2}\end{matrix}\right.\)
3, \(\left\{{}\begin{matrix}8\left(x^2+y^2\right)+4xy+\frac{5}{\left(x+y\right)^2}=13\\2x+\frac{1}{x+y}=1\end{matrix}\right.\)
Giải hệ phương trình
1. left{{}begin{matrix}x^2+y^2+2x+2yleft(x+2right)left(y+2right)left(frac{x}{y+2}right)^2+left(frac{y}{x+2}right)^21end{matrix}right.
2. left{{}begin{matrix}x^2-2xy-66y+2xfrac{3x^2}{y+1}4-xend{matrix}right.
3.left{{}begin{matrix}x^2-yy^2-xx^2-xy+3end{matrix}right.
4.left{{}begin{matrix}x+y+frac{1}{x}+frac{1}{y}frac{9}{2}xy+frac{1}{xy}+frac{x}{y}+frac{y}{x}5end{matrix}right.
6.left{{}begin{matrix}x^3left(x-yright)+x^2y^21x^2left(xy+3right)-3xy3end{matrix}right.
7.left{{...
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Giải hệ phương trình
1. \(\left\{{}\begin{matrix}x^2+y^2+2x+2y=\left(x+2\right)\left(y+2\right)\\\left(\frac{x}{y+2}\right)^2+\left(\frac{y}{x+2}\right)^2=1\end{matrix}\right.\)
2. \(\left\{{}\begin{matrix}x^2-2xy-6=6y+2x\\\frac{3x^2}{y+1}=4-x\end{matrix}\right.\)
3.\(\left\{{}\begin{matrix}x^2-y=y^2-x\\x^2-x=y+3\end{matrix}\right.\)
4.\(\left\{{}\begin{matrix}x+y+\frac{1}{x}+\frac{1}{y}=\frac{9}{2}\\xy+\frac{1}{xy}+\frac{x}{y}+\frac{y}{x}=5\end{matrix}\right.\)
6.\(\left\{{}\begin{matrix}x^3\left(x-y\right)+x^2y^2=1\\x^2\left(xy+3\right)-3xy=3\end{matrix}\right.\)
7.\(\left\{{}\begin{matrix}x^2+3y-6x=0\\9x^2-6xy^2+y^4-3y+9=0\end{matrix}\right.\)
8.\(\left\{{}\begin{matrix}x^2+y^2+xy=1\\x+y-xy=2y^2-x^2\end{matrix}\right.\)
9.\(\left\{{}\begin{matrix}8x^3-y=y^3-2x\\x^2+y^2=x+2y\end{matrix}\right.\)
10.\(\left\{{}\begin{matrix}2x^2-3xy+y^2+x-y=0\\x^2+x+1=y^2\end{matrix}\right.\)
11.\(\left\{{}\begin{matrix}\left(x^2+y^2\right)\left(x+y+2\right)=4\left(y+2\right)\\x^2+y^2+\left(y+2\right)\left(x+y+2\right)=4\left(y+2\right)\end{matrix}\right.\)
12. \(\left\{{}\begin{matrix}x^2+7=4y^2+4y\\x^2+3xy+2y^2+x+y=0\end{matrix}\right.\)
13. \(\left\{{}\begin{matrix}x^2+y^2=5\\x^3+2y^3+\left(x-5\right)^2+\left(y+5\right)^2=55\end{matrix}\right.\)
14. \(\left\{{}\begin{matrix}\frac{1}{x^2}+\frac{1}{y^2}=3+x^2y^2\\\frac{1}{x^3}+\frac{1}{y^3}+3=x^3y^3\end{matrix}\right.\)
15.\(\left\{{}\begin{matrix}x^2+y^2+4x+2y=3\\x^2+7y^2-4xy+6y=13\end{matrix}\right.\)
16. \(\left\{{}\begin{matrix}x^2-5xy+x-5y^2=42\\7xy+6y^2+42=x\end{matrix}\right.\)
17.\(\left\{{}\begin{matrix}x^2+xy+y^2=13\\x^4+x^2y^2+y^4=91\end{matrix}\right.\)
18.\(\left\{{}\begin{matrix}x^2=\left(2-y\right)\left(2+y\right)\\2x^3=\left(x+y\right)\left(4-xy\right)\end{matrix}\right.\)
Đây là các bài hệ trong đề thi chuyên toán mong mọi người giúp vì mình bận quá nên không thể làm hết được ạ