Question

Cho \(x^2y-xy^2+x^2z-xz^2+y^2z+yz^2=2xyz\). CMR: trong 3 số \(x,y,z\) có ít nhất hai số bằng nhau hoặc đối nhau.

Answer 1

\(x^2y-xy^2+x^2z-xz^2+y^2z+yz^2=2xyz\)

\(\Leftrightarrow\left(x^2y-xy^2\right)+\left(x^2z-xyz\right)-\left(xz^2-yz^2\right)-\left(xyz-y^2z\right)=0\)

\(\Leftrightarrow xy\left(x-y\right)+xz\left(x-y\right)-z^2\left(x-y\right)-yz\left(x-y\right)=0\)

\(\Leftrightarrow\left(x-y\right)\left(xy+xz-z^2-yz\right)=0\)

\(\Leftrightarrow\left(x-y\right)\left[x\left(y+z\right)-z\left(y+z\right)\right]=0\)

\(\Leftrightarrow\left(x-y\right)\left(x-z\right)\left(y+z\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=y\\x=z\\y=-z\end{matrix}\right.\)\(\left(đpcm\right)\)