5.
$
(2x - 6)^2 \ge 0; |y - 2012| \ge 0 \\\Rightarrow (2x - 6)^2 + |y - 2012| + 67 \ge 67 \\\Rightarrow \frac{2010}{(2x - 6)^2 + |y - 2012| + 67} \le \frac{2010}{67} = 30 \\ |A| - |B| \ge |A - B| \\ |y - 2011| - |y - 2012| \ge |y - 2011 - y + 2012| = |1| = 1 \\ \Rightarrow 31 - |y - 2011| + |y - 2012| = 31 - (|y - 2011| - |y - 2012|) \le 31 - 1 = 30 \\ VT \le 30; VP \ge 30 \Rightarrow VT = VP = 30\\ Dấu\ "=" \ khi \ : \\ \left\{\begin{matrix}
y - 2012 \ge 0\\
2x - 6 = 0\\
y - 2012 = 0
\end{matrix}\right. \\\Rightarrow x = 3; y = 2012 $