Ta có:
$x+y+z=2\sqrt{x-34}+4\sqrt{y-21}+6\sqrt{z-4}+45$
$\Leftrightarrow (x-34)-2\sqrt{x-34}+1+(y-21)-4\sqrt{y-21}+4+(z-4)-6\sqrt{z-4}+9=0$
$\Leftrightarrow (\sqrt{x-34}-1)^2+(\sqrt{y-21}-2)^2+(\sqrt{z-4}-3)^2=0$
$\Leftrightarrow \left\{\begin{array}{l}\sqrt{x-34}=1\\\sqrt{y-21}=2\\\sqrt{z-4}=3\end{array}\right.$
$\Leftrightarrow \left\{\begin{array}{l}x=35\\y=25\\z=13\end{array}\right.$
Từ đó suy ra: $T=2012$