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Vì \(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=1\)
\(\Rightarrow\left(x+y+z\right)\left(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\right)=x+y+z\)
\(\Leftrightarrow\frac{x^2}{y+z}+\frac{xy}{z+x}+\frac{zx}{x+y}+\frac{xy}{y+z}+\frac{y^2}{z+x}+\frac{yz}{x+y}+\frac{zx}{y+z}+\frac{yz}{z+x}+\frac{z^2}{x+y}=x+y+z\)
\(\Leftrightarrow\left(\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\right)+\left(\frac{xy+yz}{z+x}\right)+\left(\frac{yz+zx}{x+y}\right)+\left(\frac{zx+xy}{y+z}\right)=x+y+z\)
\(\Leftrightarrow\left(\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\right)+\frac{y\left(z+x\right)}{z+x}+\frac{z\left(x+y\right)}{x+y}+\frac{x\left(y+z\right)}{y+z}=x+y+z\)
\(\Leftrightarrow\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}+x+y+z=x+y+z\)
\(\Leftrightarrow\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=0\)
\(\Rightarrow M=2019\)
thay z = -(x+y) , y = -(z+x),... vao
=> Duoc bieu thuc trong do co 1/xy + 1/yz + 1/zx = (x+y+z)/xyz = 0
\(x+y+z=0\Rightarrow x+y=-z\)
\(\Rightarrow\left(x+y\right)^2=\left(-z\right)^2\Rightarrow x^2+2xy+y^2=z^2\Rightarrow x^2+y^2-z^2=-2xy\)
Tương tự: \(y^2+z^2-x^2=-2yz,x^2+z^2-y^2=-2xz\)
\(\frac{1}{y^2+z^2-x^2}+\frac{1}{x^2+y^2-z^2}+\frac{1}{x^2+z^2-y^2}\)
\(=\frac{1}{-2yz}+\frac{1}{-2xy}+\frac{1}{-2xz}=\frac{x+y+z}{-2xyz}=0\)
chắc câu này a đăng lên cho vui :vv
Ta có : \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2< =>\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2=2^2=4\)
\(< =>\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2}{xy}+\frac{2}{yz}+\frac{2}{zx}=4\)
\(< =>\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}-\left(\frac{2}{xy}-\frac{1}{z^2}\right)+\frac{2}{xy}+\frac{2}{yz}+\frac{2}{zx}+4=4\)
\(< =>\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}-\frac{2}{xy}+\frac{1}{z^2}+\frac{2}{xy}+\frac{2}{yz}+\frac{2}{zx}=4-4\)
\(< =>\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{1}{z^2}+\frac{2}{yz}+\frac{2}{zx}=0\)
\(< =>\left(\frac{1}{x^2}+\frac{2}{zx}+\frac{1}{z^2}\right)+\left(\frac{1}{y^2}+\frac{2}{yz}+\frac{1}{z^2}\right)=0\)
\(< =>\left(\frac{1}{x}+\frac{1}{z}\right)^2+\left(\frac{1}{y}+\frac{1}{z}\right)^2=0< =>\frac{1}{x}=\frac{1}{y}=-\frac{1}{z}\)
\(< =>x=y=-z\)Thế vào giả thiết ta được : \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2\)
\(< =>\frac{1}{-z}+\frac{1}{-z}+\frac{1}{z}=2< =>\frac{-1}{z}+\frac{-1}{z}+\frac{1}{z}=2\)
\(< =>\frac{-1-1+1}{z}=2< =>2z=-1< =>z=-\frac{1}{2}\)
Suy ra \(x=y=-z=-\left(-\frac{1}{2}\right)=\frac{1}{2}< =>\hept{\begin{cases}x=\frac{1}{2}\\y=\frac{1}{2}\\z=-\frac{1}{2}\end{cases}}\)
Nên \(P=\left(x+2y+z\right)^{2019}=\left(\frac{1}{2}+2.\frac{1}{2}-\frac{1}{2}\right)^{2019}=1^{2019}=1\)
\(\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=1\)
\(\Rightarrow\frac{x\left(x+y+z\right)}{y+z}+\frac{y\left(x+y+z\right)}{x+z}+\frac{z\left(x+y+z\right)}{x+y}=x+y+z\)
\(\Rightarrow\frac{x^2}{y+z}+x+\frac{y^2}{x+z}+y+\frac{z^2}{x+y}+z=x+y+z\)
\(\Rightarrow\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=0\)
\(\Rightarrow M=2019+0=2019\)