Có: \(\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=1\)
⇒(x+y+z)(\(\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}\))=x+y+z
⇔\(\frac{x^2+xy+xz}{y+z}+\frac{xy+y^2+yz}{x+z}+\frac{xz+yz+z^2}{x+y}=x+y+z\)
⇔\(\frac{x^2}{y+z}+\frac{x\left(y+z\right)}{y+z}+\frac{y^2}{x+z}+\frac{y\left(x+z\right)}{x+z}+\frac{z^2}{x+y}+\frac{z\left(x+y\right)}{x+y}=x+y+z\)
⇔\(\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}+x+y+z=x+y+z\)
Hay M+x+y+z=x+y+z
=>M=0
Lời giải:
Từ \(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=1\)
\(\Rightarrow \left\{\begin{matrix} \frac{x^2}{y+z}+\frac{xy}{z+x}+\frac{xz}{x+y}=x\\ \frac{xy}{y+z}+\frac{y^2}{z+x}+\frac{zy}{x+y}=y\\ \frac{xz}{y+z}+\frac{yz}{z+x}+\frac{z^2}{x+y}=z\end{matrix}\right.\)
Cộng theo vế cả 3 đẳng thức trên:
\(\Rightarrow \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}+\frac{xy+yz}{x+z}+\frac{xz+yz}{x+y}+\frac{xy+xz}{y+z}=x+y+z\)
\(\Leftrightarrow \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}+y+z+x=x+y+z\)
\(\Leftrightarrow \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=0\)
Vậy $M=0$