Part 10

Since $2016 = 2^{5}*3^{2}*7$ the equation can be written as $\sqrt{x} + \sqrt{y} = 12\sqrt{14}$ .

With $x = 14n^{2}, y = 14m^{2}$ for integers $0 \lt m \lt n$ the equation becomes

$n\sqrt{14} + m\sqrt{14} = 12\sqrt{14} \Longrightarrow n + m = 12$ .

This gives us the possible pairings $(n,m) = (11,1), (10,2), (9,3), (8,4)$ and $(7,5)$ , the only one of which has $m$ not dividing $n$ being $(7,5)$ . Thus $x + y = 14*(7^{2} + 5^{2}) = \boxed{1036}$ .

@Gurīdo Cuong