a)Ta có : \(A=x^2+4y^2-4x+32y+2078\)

\(\Leftrightarrow\left(x^2-4x+4\right)+\left(\left(2y\right)^2+32y+8^2\right)+2010\)

\(\Leftrightarrow\left(x-2\right)^2+\left(2y+8\right)^2+2010\)

Mà \(\left(x-2\right)^2\ge0\) với mọi x
\(\left(2y+8\right)^2\ge0\) với mọi y
\(\Rightarrow\left(x-2\right)^2+\left(2y+8\right)^2+2010\ge2010\) với mọi x,y
Dấu bằng xảy ra khi\(x-2=0\Rightarrow x=2\)
\(2y+8=0\Rightarrow y=-4\)
b)

Ta có:\(9x^2+y^2+2z^2-18x+4z-6y+20=0\)

\(9\left(\right.\) \(x^2-2x+1)+\) \(\left(\right.\) \(\left(y\right)^2-6y+\left(3\right)^2\) \()+2\) \(\left(\right.\) \(\left(z\right)^2+2z+1)=0\) \(\)

\(\Leftrightarrow9\left(x-1\right)^2+\left(y+3\right)^2+\left(z+1\right)^2=0\)
Mà\(\left(x-1\right)^2\ge0\)

\(\left(y+3\right)^2\ge0\)

\(\left(z+1\right)^2\ge0\)
\(\Rightarrow\) Dấu bằng xảy ra khi \(x-1=0\Rightarrow x=1\)

\(y+3=0\Rightarrow y=-3\)

\(z+1=0\Rightarrow z=-1\)

\(a)x^3+y^3+z^3-3xyz=0\)

\(\Leftrightarrow x^3+y^3+3x^2y+3xy^2-3x^2y-3xy^2+z^3-3xyz=0\)

\(\) \(\Leftrightarrow\left(x+y\right)^3+z^3-3xy\left(x+y+z\right)=0\)

\(\Leftrightarrow\left(x+y+z\right)\left(\right.\) \(\left(x+y\right)^2-z\left(x+y\right)+z^2-3xy)=0\)

\(\Leftrightarrow\left(x+y+z\right)\left(\right.\) \(x^2+2xy+y^2-xz-yz+z^2-3xy)=0\)

Mà \(x+y+z=0\)
\(\Rightarrow0=0\left(đpcm)\right.\)

\(b)\left(x^2y^2+y^2z^2+x^2z^2+2\left.x^2yz+2xy^2z+2xyz^2\right)\right.=x^2y^2+y^2z^2+x^2z^2\)

\(\Leftrightarrow2\left(\right.\) \(x^2yz+xy^2z+xyz^2)=0\)

\(\Leftrightarrow2\left(x+y+z\right)\left(xyz\right)=0\)
Mà \(x+y+z=0\)

\(\Rightarrow0=0\left(đpcm\right)\)

\(c)\) Ta có:\(x+y+z=0\)

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

\(\Rightarrow x^2+y^2+z^2+2\left(\right.\) \(x^2yz+xy^2z+xyz^2)=0\)

\(\Rightarrow2\left(\right.\) \(xy+yz+xz^{})=-\left(\right.\) \(x^2+y^2+z^2)\)

\(\Rightarrow4\left(\right.\) \(xy+yz+xz)^2=\) \(x^4+y^4+z^4+2\left(\right.\) \(x^2y^2+y^2z^2+x^2z^2)\left(1\right)\)

Mà ta có: \(\left(xy+yz+xz\right)^2=x^2y^2+y^2z^2+x^2z^2\) (theo câu b)

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

\(\left(1\right)-\left(2\right)\Leftrightarrow2\left(xy+yz+xz\right)^2=x^4+y^4+z^4\left(đpcm\right)\)