Các câu hỏi tương tự
đặt Afrac{sqrt{yz}}{x+3sqrt{yz}}+frac{sqrt{zx}}{y+3sqrt{zx}}+frac{sqrt{xy}}{z+3sqrt{xy}}Rightarrow1-3Afrac{x}{x+3sqrt{yz}}+frac{y}{y+3sqrt{zx}}+frac{z}{z+3sqrt{xy}}gefrac{x}{x+frac{3}{2}left(y+zright)}+frac{y}{y+frac{3}{2}left(z+xright)}+frac{z}{z+frac{3}{2}left(x+yright)}frac{2x}{2x+3left(y+zright)}+frac{2y}{2y+3left(z+xright)}+frac{2z}{2z+3left(x+yright)}frac{2x^2}{2x^2+3xy+3xz}+frac{2y^2}{2y^2+3yz+3xy}+frac{2z^2}{2z^2+3zx+3yz}gefrac{2left(x+y+zright)^2}{2left(x^2+y^2+z^2right)+6left(xy+yz+zxr...
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đặt \(A=\frac{\sqrt{yz}}{x+3\sqrt{yz}}+\frac{\sqrt{zx}}{y+3\sqrt{zx}}+\frac{\sqrt{xy}}{z+3\sqrt{xy}}\)
\(\Rightarrow1-3A=\frac{x}{x+3\sqrt{yz}}+\frac{y}{y+3\sqrt{zx}}+\frac{z}{z+3\sqrt{xy}}\)
\(\ge\frac{x}{x+\frac{3}{2}\left(y+z\right)}+\frac{y}{y+\frac{3}{2}\left(z+x\right)}+\frac{z}{z+\frac{3}{2}\left(x+y\right)}\)
\(=\frac{2x}{2x+3\left(y+z\right)}+\frac{2y}{2y+3\left(z+x\right)}+\frac{2z}{2z+3\left(x+y\right)}\)
\(=\frac{2x^2}{2x^2+3xy+3xz}+\frac{2y^2}{2y^2+3yz+3xy}+\frac{2z^2}{2z^2+3zx+3yz}\)
\(\ge\frac{2\left(x+y+z\right)^2}{2\left(x^2+y^2+z^2\right)+6\left(xy+yz+zx\right)}=\frac{2\left(x+y+z\right)^2}{2\left(x+y+z\right)^2+2\left(xy+yz+zx\right)}\)
\(\ge\frac{2\left(x+y+z\right)^2}{2\left(x+y+z\right)^2+\frac{2}{3}\left(x+y+z\right)^2}=\frac{2\left(x+y+z\right)^2}{\frac{8}{3}\left(x+y+z\right)^2}=\frac{3}{4}\)
\(\Rightarrow1-3A\ge\frac{3}{4}\Rightarrow A\le\frac{3}{4}\left(Q.E.D\right)\)
ta có:(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)-abcgeleft(a+b+cright)left(ab+bc+caright)-frac{1}{9}left(a+b+cright)left(ab+bc+caright)frac{8}{9}left(a+b+cright)left(ab+bc+caright)frac{x}{x+yz}+frac{y}{y+zx}+frac{z}{z+xy}frac{x}{left(x+yright)left(x+zright)}+frac{y}{left(y+xright)left(y+zright)}+frac{z}{left(z+xright)left(z+yright)}frac{2left(xy+yz+zxright)}{left(x+yright)left(y+zright)left(z+xright)}lefrac{9}{4left(xy+yz+zxright)}frac{9}{4}
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ta có:
(a+b)(b+c)(c+a)=(a+b+c)(ab+bc+ca)-abc\(\ge\left(a+b+c\right)\left(ab+bc+ca\right)-\frac{1}{9}\left(a+b+c\right)\left(ab+bc+ca\right)=\frac{8}{9}\left(a+b+c\right)\left(ab+bc+ca\right)\)
\(\frac{x}{x+yz}+\frac{y}{y+zx}+\frac{z}{z+xy}=\frac{x}{\left(x+y\right)\left(x+z\right)}+\frac{y}{\left(y+x\right)\left(y+z\right)}+\frac{z}{\left(z+x\right)\left(z+y\right)}=\frac{2\left(xy+yz+zx\right)}{\left(x+y\right)\left(y+z\right)\left(z+x\right)}\le\frac{9}{4\left(xy+yz+zx\right)}=\frac{9}{4}\)
câu 1: giải hệ phương trìnhleft(x+yright)^2+left(y+zright)^4+....+left(x+zright)^{100}-left(y+z+xright)left(xyright)^2+2left(yzright)^4+....+100left(zxright)^{100}-[left(x+y+zright)+2left(yz+zx+xyright)+......+99left(x+y+zright)]left(frac{1}{x}+frac{1}{y}right)^2+left(frac{1}{y^2}+frac{1}{z^2}right)^2+...+left(frac{1}{x^{99}}+frac{1}{z^{99}}right)^2-frac{1}{left(xyright)^2+2left(yzright)^2+.....+99left(zxright)^2}tìm x,y,z
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câu 1: giải hệ phương trình
\(\left(x+y\right)^2+\left(y+z\right)^4+....+\left(x+z\right)^{100}=-\left(y+z+x\right)\)
\(\left(xy\right)^2+2\left(yz\right)^4+....+100\left(zx\right)^{100}=-[\left(x+y+z\right)+2\left(yz+zx+xy\right)+......+99\left(x+y+z\right)]\)\(\left(\frac{1}{x}+\frac{1}{y}\right)^2+\left(\frac{1}{y^2}+\frac{1}{z^2}\right)^2+...+\left(\frac{1}{x^{99}}+\frac{1}{z^{99}}\right)^2=-\frac{1}{\left(xy\right)^2+2\left(yz\right)^2+.....+99\left(zx\right)^2}\)
tìm x,y,z
Cho x>0,y<0 và x+y=1/ Rút gọn biểu thức:
\(A=\frac{y-x}{xy}:\left[\frac{y^2}{\left(x-y\right)^2}-\frac{2x^2y}{\left(x^2-y^2\right)^2}+\frac{x^2}{x^2-y^2}\right]\)
Chứng minh rằng A<-4
Cho \(\hept{\begin{cases}x,y,z>0\\xy+yz+zx=1\end{cases}}\). Chứng minh rằng:
\(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\ge3+\sqrt{\frac{\left(x+y\right)\left(x+z\right)}{x^2}}+\sqrt{\frac{\left(y+z\right)\left(y+x\right)}{y^2}}+\sqrt{\frac{\left(z+x\right)\left(z+y\right)}{z^2}}\)
cho x;y;z0xy+yz+xzxyzvà left(x+yright)left(frac{1}{z}+frac{1}{xy}right)+left(y+zright)left(frac{1}{x}+frac{1}{yz}right)+left(x+zright)left(frac{1}{y}+frac{1}{xz}right)1tính giá trị của biểu thứcAsqrt{frac{left(2x+yzright)left(2y+xzright)}{left(y+zright)left(x+zright)}}+sqrt{frac{left(2y+xzright)left(2z+xyright)}{left(x+zright)left(x+yright)}}+sqrt{frac{left(2z+xyright)left(2x+yzright)}{left(x+yright)left(y+zright)}}
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cho \(x;y;z>0\)
\(xy+yz+xz=xyz\)
và \(\left(x+y\right)\left(\frac{1}{z}+\frac{1}{xy}\right)+\left(y+z\right)\left(\frac{1}{x}+\frac{1}{yz}\right)+\left(x+z\right)\left(\frac{1}{y}+\frac{1}{xz}\right)=1\)
tính giá trị của biểu thức
\(A=\sqrt{\frac{\left(2x+yz\right)\left(2y+xz\right)}{\left(y+z\right)\left(x+z\right)}}+\sqrt{\frac{\left(2y+xz\right)\left(2z+xy\right)}{\left(x+z\right)\left(x+y\right)}}+\sqrt{\frac{\left(2z+xy\right)\left(2x+yz\right)}{\left(x+y\right)\left(y+z\right)}}\)
1.Giải hệ pt1)hept{begin{cases}frac{1}{x}+frac{1}{y}+frac{1}{z}3xy+yz+zx3frac{1}{1+x+xy}+frac{1}{1+y+yz}+frac{1}{1+z+zx}xend{cases}}2)hept{begin{cases}xy+yz+zx3left(x+yright)left(y+zright)sqrt{3}zleft(1+y^2right)left(y+zright)left(z+xright)sqrt{3}xleft(1+z^2right)end{cases}}3)hept{begin{cases}xy+yz+zx31+x^2left(y+zright)+xyz4y1+y^2left(z+xright)+xyz4zend{cases}}
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1.Giải hệ pt
1)\(\hept{\begin{cases}\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=3\\xy+yz+zx=3\\\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}=x\end{cases}}\)
2)\(\hept{\begin{cases}xy+yz+zx=3\\\left(x+y\right)\left(y+z\right)=\sqrt{3}z\left(1+y^2\right)\\\left(y+z\right)\left(z+x\right)=\sqrt{3}x\left(1+z^2\right)\end{cases}}\)
3)\(\hept{\begin{cases}xy+yz+zx=3\\1+x^2\left(y+z\right)+xyz=4y\\1+y^2\left(z+x\right)+xyz=4z\end{cases}}\)
Cho 3 số thực x, y, z thỏa mãn: \(x+y+z\le\frac{3}{2}\). Tìm Min \(P=\frac{x\left(yz+1\right)^2}{z^2\left(zx+1\right)}+\frac{y\left(xz+1\right)^2}{x^2\left(xy+1\right)}+\frac{z\left(xy+1\right)^2}{y^2\left(yz+1\right)}\)
cho x+y+z<\(\frac{3}{2}\)
tìm min p=\(\frac{x\left(yz+1\right)^2}{z^2\left(zx+1\right)}+\frac{y\left(zx+1\right)^2}{x^2\left(xy+1\right)}+\frac{z\left(xy+1\right)}{y^2\left(yz+1\right)}^2\)