\(\left(x+y+1\right)^2=3\left(x^2+y^2+1\right)\)
\(\Leftrightarrow x^2+y^2+1+2xy+2x+2y=3x^2+3y^2+3\)
\(\Leftrightarrow2x^2+2y^2-2xy-2x-2y+2=0\)
\(\Leftrightarrow\left(x^2-2xy+y^2\right)+\left(x^2-2x+1\right)+\left(y^2-2y+1\right)=0\)
\(\Leftrightarrow\left(x-y\right)^2+\left(x-1\right)^2+\left(y-1\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\x-1=0\\y-1=0\end{matrix}\right.\) \(\Rightarrow x=y=1\)
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