Question

[Thiên Ân]

Cho  \(x^2+y^2+z^2=xy+yz+zx\)   và   \(x^{2016}+y^{2016}+z^{2016}=3^{2017}\)

Tính   \(x,y,z\)

Answer 1

[Lê Hồ Trọng Tín]

Ta có: x²+y²+z²=xy+yz+zx (gt)

\(\Leftrightarrow\)2x²+2y²+2z²=2xy+2yz+2zx

\(\Leftrightarrow\)x²-2xy+y²+y²-2yz+z²+z²-2zx+x²=0

\(\Leftrightarrow\)(x-y)²+(y-z)²+(z-x)²=0

\(\Leftrightarrow\)x=y,y=z,z=x

\(\Leftrightarrow\)x=y=z

Khi đó:x²⁰¹⁶+y²⁰¹⁶+z²⁰¹⁶=3²⁰¹⁷

\(\Leftrightarrow\)3.x²⁰¹⁶=3²⁰¹⁷

\(\Leftrightarrow\)x²⁰¹⁶=3²⁰¹⁶

\(\Leftrightarrow\)x=\(\pm\)3

Vậy:x=y=z=3 hoặc x=y=z=-3

Answer 2

[Thanh Tùng DZ]

Ta có : \(x^2+y^2+z^2=xy+yz+xz\Leftrightarrow x^2+y^2+z^2-xy-yz-xz=0\)

\(\Leftrightarrow2\left(x^2+y^2+z^2-xy-yz-xz\right)=0\)

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

\(\Leftrightarrow x=y=z\)

Mà \(x^{2016}+y^{2016}+z^{2016}=3^{2017}\)

\(x^{2016}=y^{2016}=z^{2016}=\frac{3^{2017}}{3}=3^{2016}\)

\(\Rightarrow x=y=z=\sqrt[2016]{3^{2016}}=3\)