Question

cho x,y,z dương thoả mãn điều kiện : (x+y)(y+z)(z+x)=8xyz.

CMR x=y=z

Answer 1

(x + y)(y + z)(x + z) = 8xyz

⇒ (xy + xz + y² + yz)(x + z) - 8xyz = 0

⇒ x²y + xyz + x²z + xz² + y²x + y²z + xyz + yz² - 8xyz = 0

⇒ x²y - 2xyz + yz² + xy² - 2xyz + xz² + x²z - 2xyz + y²z = 0

⇒ y(x - z)² + x(y - z)² + z(x - y)² = 0

mà x, y, z > 0 (gt)

⇒ \(\left\{{}\begin{matrix}\left(x-z\right)^2=0\\\left(y-z\right)^2=0\\\left(x-y\right)^2=0\end{matrix}\right.\)

⇒ \(\left\{{}\begin{matrix}x-z=0\\y-z=0\\x-y=0\end{matrix}\right.\)

⇒ \(\left\{{}\begin{matrix}x=z\\y=z\\x=y\end{matrix}\right.\)

⇒ x = y = z