\(x^2-2xy+6y^2-12x+2y+41=0\\ \Leftrightarrow\left[\left(x^2-2xy+y^2\right)-12\left(x-y\right)+36\right]+\left(5y^2-10y+5\right)=0\\ \Leftrightarrow\left[\left(x-y\right)^2-12\left(x-y\right)+36\right]+5\left(y^2-2y+1\right)=0\\ \Leftrightarrow\left(x-y-6\right)^2+5\left(y-1\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}x=y+6\\y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=7\\y=1\end{matrix}\right.\)
\(\Leftrightarrow A=\dfrac{2020-2019\left(9-7-1\right)^{2019}-\left(7-6\cdot1\right)^{2018}}{1^{1010}}\\ \Leftrightarrow A=2020-2019\cdot1^{2019}-1^{2018}=2020-2019-1=0\)
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