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Từ \(3\left(x^2+y^2+z^2\right)=\left(x+y+z\right)^2\)
Suy ra: x=y=z
\(\Rightarrow3x^{2018}=3y^{2018}=3z^{2018}=27^{673}=3^{2019}\)
\(\Leftrightarrow x^{2018}=y^{2018}=z^{2018}=3^{2018}\)
\(\Rightarrow x,y,z=3\)
Dễ tính A
mình sửa đề nhé~
Có: \(\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\ge0\forall x;y;z\)
\(\Rightarrow2.\left(x^2+y^2+z^2\right)-2xy-2yz-2xz\ge0\forall x;y;z\)
\(\Leftrightarrow2.\left(x^2+y^2+z^2\right)\ge2xy+2yz+2xz\forall x;y;z\)
\(\Leftrightarrow3.\left(x^2+y^2+z^2\right)\ge x^2+y^2+z^2+2xy+2yz+2xz\forall x;y;z\)
\(\Leftrightarrow3.\left(x^2+y^2+z^2\right)\ge\left(x+y+z\right)^2\forall x;y;z\)
Mà \(3.\left(x^2+y^2+z^2\right)=\left(x+y+z\right)^2\)
\(\left\{{}\begin{matrix}\left(x-y\right)^2=0\\\left(y-z\right)^2=0\\\left(z-x\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\y-z=0\\z-x=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=y\\y=z\\x=z\end{matrix}\right.\Leftrightarrow x=y=z\)
Có: \(x^{2018}+y^{2018}+z^{2018}=27^{673}\)
\(\Leftrightarrow3.x^{2018}=27^{673}\)
\(\Leftrightarrow x^{2018}=3^{2018}\)
\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=-3\end{matrix}\right.\)
đến đây bạn tự làm nốt nhé
\(A=\frac{x^2}{\left(x-y\right)\left(x-z\right)}+\frac{y^2}{\left(y-x\right)\left(y-z\right)}+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)
\(=\frac{x^2}{\left(x-y\right)\left(x-z\right)}-\frac{y^2}{\left(x-y\right)\left(y-z\right)}+\frac{z^2}{\left(x-z\right)\left(y-z\right)}\)
\(=\frac{x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left(x-y\right)\)
\(=x^2y-x^2z-xy^2+y^2z+z^2\left(x-y\right)\)
\(=xy\left(x-y\right)-z\left(x-y\right)\left(x+y\right)+z^2\left(x-y\right)\)
\(=\left(x-y\right)\left[xy-zx-zy+z^2\right]\)
\(=\left(x-y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]=\left(x-y\right)\left(x-z\right)\left(y-z\right)\)
Vậy A = 1
2x2 + 2y2 + 3xy - x + y + 1 = 0
2x2 + 2y2 + 4xy - xy - x + y + 1 = 0
(2x2 + 2y2 + 4xy) + (-xy - x) + (y + 1) = 0
2(x + y)2 - x(y + 1) + (y + 1) = 0
2(x + y)2 + (y + 1)(1 - x) = 0
Do (x + y)2 \(\ge0\)
\(\Rightarrow\) 2(x + y)2 \(\ge0\)
\(\Rightarrow\) 2(x + y)2 + (y + 1)(1 - x) = 0 \(\Leftrightarrow\) (y + 1)(1 - x) = 0
\(\Rightarrow y+1=0;1-x=0\)
*) y + 1 = 0
y = -1
*) 1 - x = 0
x = 1
Với x = 1; y = -1, ta có:
B = [1 + (-1)]2018 + (1 - 2)2018 + (-1 - 1)2018
= 1 + 22018
Ta có : \(3\left(x^2+y^2+z^2\right)=\left(x+y+z\right)^2\)
\(\Leftrightarrow3\left(x^2+y^2+z^2\right)=x^2+y^2+z^2+2\left(xy+yz+zx\right)\)
\(\Leftrightarrow2\left(x^2+y^2+z^2-xy-yz-zx\right)=0\)
\(\Leftrightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\)
\(\Leftrightarrow x=y=z\)
Khi đó : \(3x^{2018}=27^{673}=\left(3^3\right)^{673}=3^{2019}\)
\(\Leftrightarrow x^{2018}=3^{2018}\)
\(\Leftrightarrow\orbr{\begin{cases}x=y=z=3\\x=y=z=-3\end{cases}}\)
Đến đây tự tính A nha!