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Lời giải:
$x^3+y^3+z^3-3xyz=0$
$\Leftrightarrow (x+y)^3-3xy(x+y)+z^3-3xyz=0$
$\Leftrightarrow (x+y)^3+z^3-3xy(x+y+z)=0$
$\Leftrightarrow (x+y+z)[(x+y)^2-z(x+y)+z^2]-3xy(x+y+z)=0$
$\Leftrightarrow (x+y+z)(x^2+y^2+z^2-xy-yz-xz)=0$
Đến đây xét 2TH:
TH1: $x+y+z=0$
\(\Rightarrow \left\{\begin{matrix} x+y=-z\\ y+z=-x\\ x+z=-y\end{matrix}\right.\)
\(\Rightarrow B=-16+(-3)+(-2038)=-2057\)
TH2: $x^2+y^2+z^2-xy-yz-xz=0$
$\Leftrightarrow \frac{(x-y)^2+(y-z)^2+(z-x)^2}{2}=0$
$\Rightarrow (x-y)^2=(y-z)^2=(z-x)^2=0$
$\Rightarrow x=y=z$ (vô lý vì $x,y,z$ đôi một khác nhau)
Vậy.......
\(x^3+y^3+z^3-3xyz=0\Leftrightarrow x^3+y^3+3xy\left(x+y\right)+z^3-3xy\left(x+y\right)-3xyz=0\)
\(\Leftrightarrow\left(x+y\right)^3+z^3-3xy\left(x+y+z\right)=0\)
\(\Leftrightarrow\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2-3xy\right]=0\)
\(\Leftrightarrow\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)=0\)
\(\Leftrightarrow\left(x+y+z\right)\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right]=0\)
\(\Rightarrow\left[{}\begin{matrix}x+y+z=0\\x=y=z\end{matrix}\right.\)
- Nếu \(x+y+z=0\Rightarrow B=\frac{-16z}{z}-\frac{3x}{x}-\frac{2038y}{y}=...\)
- Nếu \(x=y=z\Rightarrow B=\frac{16.2z}{z}+\frac{3.2x}{x}+\frac{2038.2y}{y}=...\)
\(x^3+y^3+z^3-3xyz=0\)
\(\Leftrightarrow\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz=0\)
\(\Leftrightarrow\left(x+y+z\right)\left[\left(x+y\right)^2-z\left(x+y\right)+z^2\right]-3xy\left(x+y+z\right)=0\)
\(\Leftrightarrow\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)=0\)
\(\Leftrightarrow\dfrac{1}{2}\left(x+y+z\right)\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right]=0\)
\(\Leftrightarrow x+y+z=0\Rightarrow\left\{{}\begin{matrix}x+y=-z\\y+z=-x\\x+z=-y\end{matrix}\right.\)
\(B=\dfrac{16.\left(-z\right)}{z}+\dfrac{3.\left(-x\right)}{x}-\dfrac{2019.\left(-y\right)}{y}=2019-19=2000\)
dat a=x-y
b=y-z
c=z-x
a+b+c=0=x+y+z
\(\left(\frac{a}{z}+\frac{b}{x}+\frac{c}{y}\right)\left(\frac{z}{a}+\frac{x}{b}+\frac{y}{c}\right)\)
dung bumiakopsky de giai
...........................................
phân tích gt sau đó suy ra x+y+x=0
từ đây tính đc x+y=? y+z=? x+z=?
ta được kết quả là'; -2006
Xét \(x^3+y^3+z^3=3xyz\)
\(x^3+y^3+z^3-3xyz=0\)
\(\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz=0\)
\(\left(x+y\right)^3+z^3-3xy\left(x+y+z\right)=0\)
\(\left(x+y+z\right)\left(x^2+2xy+y^2-xy-yz+z^2\right)-3xy\left(x+y+z\right)=0\)
\(\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)=0\)
TH1:\(x+y+z=0\)
\(\Rightarrow x+y=-z;y+z=-x;z+x=-y\left(1\right)\)
Thay (1) vô pt cần tính:
\(\frac{2016xyz}{-z.-x.-y}=\frac{2016xyz}{-\left(xyz\right)}=-2016\)
TH2:\(x^2+y^2+z^2-xy-yz-xz=0\)
Nhân 2 vế với 2
\(2x^2+2y^2+2z^2-2xy-2yz-2xz=0\)
\(x^2-2xy+y^2+x^2-2xz+z^2+y^2-2yz+z^2=0\)
\(\left(x-y\right)^2+\left(x-z\right)^2+\left(y-z\right)^2=0\)
Do VT dương
\(\Rightarrow\hept{\begin{cases}\left(x-y\right)^2=0\\\left(x-z\right)^2=0\\\left(y-z\right)^2=0\end{cases}\Rightarrow}\hept{\begin{cases}x-y=0\\x-z=0\\y-z=0\end{cases}\Rightarrow}\hept{\begin{cases}x=y\\x=z\\y=z\end{cases}}\Rightarrow x=y=z\)
Thay y,z ở pt cần tính là x
\(\Rightarrow\frac{2016x.x.x}{\left(x+x\right)\left(x+x\right)\left(x+x\right)}=\frac{2016x^3}{2x.2x.2x}=\frac{2016x^3}{8x^3}=\frac{2016}{8}=252\)
Vậy pt có thể = -2016 khi x + y + z = 0
pt có thể = 252 khi \(x^2+y^2+z^2-xy-xz-yz=0\)
Áp dụng bđt côsi cho 2 số dương lần lượt ta có :
\(1+\frac{y}{x}\ge2\sqrt{\frac{y}{x}}\)
\(1+\frac{z}{y}\ge2\sqrt{\frac{z}{y}}\)
\(1+\frac{x}{z}\ge2\sqrt{\frac{x}{z}}\)
Nhân vế theo vế ta đc : \(\left(1+\frac{y}{x}\right)\left(1+\frac{z}{y}\right)\left(1+\frac{x}{z}\right)\ge8\sqrt{\frac{xyz}{xyz}}=8\)
Dấu = xảy ra khi : \(1=\frac{y}{x}\)=> x=y và \(1=\frac{z}{y}\) => z=y và \(1=\frac{x}{z}\) => x=z
=> x=y=z
Thay vào M ta được : \(M=\frac{x^2}{2x^2}+\frac{y^2}{2y^2}+\frac{z^2}{2z^2}=\frac{3}{2}\).
X3 + Y3 + Z3 = 3XYZ
<=> X3 + Y3 + Z3 - 3XYZ = 0
<=> ( X3 + Y3 ) + Z3 - 3XYZ = 0
<=> ( X + Y )3 - 3XY( X + Y ) + Z3 - 3XYZ = 0
<=> [ ( X + Y )3 + Z3 ] - 3XY( X + Y + Z ) = 0
<=> ( X + Y + Z )[ ( X + Y )2 - ( X + Y ).Z + Z2 - 3XY ] = 0
<=> ( X + Y + Z )( X2 + Y2 + Z2 - XY - YZ - XZ ) = 0
<=> \(\orbr{\begin{cases}X+Y+Z=0\\X^2+Y^2+Z^2-XY-YZ-XZ=0\end{cases}}\)
+) X + Y + Z = 0 => \(\hept{\begin{cases}X+Y=-Z\\Y+Z=-X\\X+Z=-Y\end{cases}}\)
KHI ĐÓ : \(M=\left(1+\frac{X}{Y}\right)\left(1+\frac{Y}{Z}\right)\left(1+\frac{Z}{X}\right)=\left(\frac{X+Y}{Y}\right)\left(\frac{Y+Z}{Z}\right)\left(\frac{X+Z}{X}\right)=\frac{-Z}{Y}\cdot\frac{-X}{Z}\cdot\frac{-Y}{X}=-1\)
+) X2 + Y2 + Z2 - XY - YZ - XZ = 0
<=> 2( X2 + Y2 + Z2 - XY - YZ - XZ ) = 0
<=> 2X2 + 2Y2 + 2Z2 - 2XY - 2YZ - 2XZ = 0
<=> ( X2 - 2XY + Y2 ) + ( Y2 - 2YZ + Z2 ) + ( X2 - 2XZ + Z2 ) = 0
<=> ( X - Y )2 + ( Y - Z )2 + ( X - Z )2 = 0 (1)
DỄ DÀNG CHỨNG MINH (1) ≥ 0 ∀ X,Y,Z
DẤU "=" XẢY RA <=> X = Y = Z
KHI ĐÓ : \(M=\left(1+\frac{X}{Y}\right)\left(1+\frac{Y}{Z}\right)\left(1+\frac{Z}{X}\right)=\left(1+\frac{Y}{Y}\right)\left(1+\frac{Z}{Z}\right)\left(1+\frac{X}{X}\right)=2\cdot2\cdot2=8\)
\(x^3+y^3+z^3=3xyz\)
\(x^3+y^3+z^3-3xyz=0\)
\(\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)=0\)
\(x^2+y^2+z^2-xy-xz-yz=0\left(x+y+z\ne0\right)\)
\(2\times\left(x^2+y^2+z^2-xy-xz-yz\right)=0\times2\)
\(2x^2+2y^2+2z^2-2xy-2xz-2yz=0\)
\(x^2-2xy+y^2+x^2-2xz+z^2+y^2-2yz+z^2=0\)
\(\left(x-y\right)^2+\left(x-z\right)^2+\left(y-z\right)^2=0\)
\(\left[\begin{array}{nghiempt}x-y=0\\x-z=0\\y-z=0\end{array}\right.\)
\(\left[\begin{array}{nghiempt}x=y\\x=z\\y=z\end{array}\right.\)
x = y = z
\(P=\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)\left(1+\frac{x}{z}\right)\)
\(=\left(1+\frac{x}{x}\right)\left(1+\frac{y}{y}\right)\left(1+\frac{z}{z}\right)\)
\(=\left(1+1\right)\left(1+1\right)\left(1+1\right)\)
\(=2^3\)
\(=8\)
(x+y)^3 - 3xy(x+y) + z^3 - 3xyz = 0
(x+y+z) ( (x+y)^2 +z^2 -z(x+y) -3xy) =0
(x+y+z) ( x^2+ 2xy+y^2 +z^2- zx-zy-3xy)=0
(x+y+z) ( x^2+y^2+z^2 -zx-zy -xy)=0
Suy ra x+y+z =0
x+y = -z
y+z = -x
x+z = -y
B = -16 + (-3) +2038 = 2019
Ta có: \(x^3+y^3+z^3-3xyz=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)\)
\(\Rightarrow\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)=0\)
\(\Rightarrow\orbr{\begin{cases}x+y+z=0\\x=y=z\end{cases}}\left(x,y,z\ne0\right)\)
+) x + y + z = 0 \(\Rightarrow B=\frac{-16z}{z}+\frac{-3x}{x}-\frac{-2038y}{y}\)
\(=-16-3+2038=2019\)
+) x = y = z \(\Rightarrow B=\frac{16.2z}{z}+\frac{3.2x}{x}-\frac{2038.2y}{y}\)
\(=32+6-4076=-4038\)