Ta có \(x^2-xy=y^2-yz\Leftrightarrow x^2-y^2=xy-yz\Leftrightarrow\left(x-y\right)\left(x+y\right)=y\left(x-z\right)\)Tương tự, ta có \(\left(y-z\right)\left(y+z\right)=z\left(y-x\right),\left(z-x\right)\left(z+x\right)=x\left(z-y\right)\)
Nhân 3 vế với nhau, ta được \(\left(x+y\right)\left(y+z\right)\left(z+x\right)\left(x-y\right)\left(y-z\right)\left(z-x\right)=-xyz\left(x-y\right)\left(y-z\right)\left(z-x\right)\Leftrightarrow\left(x+y\right)\left(y+z\right)\left(z+x\right)=-xyz\Leftrightarrow x^2y+xy^2+yz^2+y^2z+zx^2+z^2x+3xyz=0\Leftrightarrow\left(x+y+z\right)\left(xy+yz+zx\right)=0\)-Xét x + y + z = 0 => tm
-Xét xy + yz + zx = 0, ta có đpcm
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