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\(x;y;z\ne0\). Giả thiết của đề bài:
\(\frac{xy}{x+y}=\frac{yz}{y+z}=\frac{xz}{z+x}\Leftrightarrow\frac{x+y}{xy}=\frac{y+z}{yz}=\frac{x+z}{xz}\Leftrightarrow\frac{1}{x}+\frac{1}{y}=\frac{1}{y}+\frac{1}{z}=\frac{1}{x}+\frac{1}{z}\Leftrightarrow\frac{1}{x}=\frac{1}{y}=\frac{1}{z}.\)
=> x = y = z
Do đó, M = 1.
Ta có \(\frac{xy}{x+y}=\frac{yz}{y+z}=\frac{xz}{x+z}\)
=> \(\frac{xyz}{xz+yz}=\frac{xyz}{xy+xz}=\frac{xyz}{xy+yz}\)
=> \(xz+yz=xy+xz=xy+yz\)(vì x ; y ;z \(\ne0\Leftrightarrow xyz\ne0\))
=> \(\hept{\begin{cases}xz+yz=xy+xz\\xy+xz=xy+yz\\xz+yz=xy+yz\end{cases}}\Rightarrow\hept{\begin{cases}yz=xy\\xz=yz\\xz=xy\end{cases}}\Rightarrow\hept{\begin{cases}z=x\\x=y\\y=z\end{cases}}\Rightarrow x=y=z\)
Khi đó M = \(\frac{x^2+y^2+z^2}{xy+yz+zx}=\frac{x^2+y^2+z^2}{x^2+y^2+z^2}=1\left(\text{vì }x=y=z\right)\)
\(\frac{2013x}{xy+2013x+2013}+\frac{y}{yz+y+2013}+\frac{z}{xz+z+1}\)
\(=\frac{x^2yz}{xy+x^2yz+xyz}+\frac{y}{yz+y+xyz}+\frac{z}{xz+z+1}\)
\(=\frac{xz}{1+xz+z}+\frac{1}{z+1+xz}+\frac{z}{xz+z+1}\)
\(=\frac{xz+z+1}{xz+z+1}=1\)
=>đpcm
2013x/xy+2013x+2013 + y/yz+y+2013 + z/xz+z+1
= xyz.x/xy+xyz.x+xyz + y/yz+y+xyz + z/xz+z+1
= xz/1+xz+z + 1/z+1+xz + z/xz+z+1
= xz+1+x/1+xz+x = 1 (đpcm)
Ta có:\(\frac{xy}{x+y}=\frac{yz}{y+z}\Rightarrow xy\left(y+z\right)=yz\left(x+y\right)\Leftrightarrow xy^2+xyz=xyz+y^2z\Leftrightarrow xy^2=y^2z\Rightarrow x=z\)(1)
\(\frac{yz}{y+z}=\frac{xz}{x+z}\Rightarrow yz\left(x+z\right)=xz\left(y+z\right)\Leftrightarrow xyz+yz^2=xyz+xz^2\Leftrightarrow yz^2=xz^2\Rightarrow y=x\)(2)
Từ (1)và(2)suy ra:x=y=z
\(\Rightarrow x^2=xy,y^2=yz,z^2=xz\)
\(\Rightarrow M=\frac{xy+yz+xz}{xy+yz+xz}=1\)
Vậy M=1
\(\frac{x}{y}=\frac{y}{z}=\frac{z}{x}=\frac{x+y+z}{y+z+x}=1\Leftrightarrow x=y=z\)
M =\(\frac{y^{670.3}}{y^{2012}}=\frac{y^{2010}}{y^{2012}}=\frac{1}{y^2}\)
Đề sai nhé mẫu mũ 2010 => M =1 mới đúng