I doubt whether the standard solution for problem E is correct.
The problem is: when two center's distance is smaller than 2R, the two circles may still be separated on the grid matrix.
Consider the following case:
Circle 1: center (0,0) radius 14
Circle 2: center (20,19) radius 14
The eculid distance between center is 
, which is less than 28.
However, the 2 circles do not intersect on a grid matrix, since no integer coodinate falls on their intersection area. So the 2 circles are still valid solution.
But it seems that the standard solution did not consider this special case. I (and many others) got wa on test 20. After I changed the intersect checking to "center distance<=2R" it got accepted.
solution: http://codeforces.com/contest/363/submission/5067126 notice the quoted code between line 77 and 82. after removing these codes the solution got WA.
A possibly wrong test case for the std solution: http://paste.ubuntu.com/6399141/
Thanks for your post. I also have the same opinion with you on this issue (but it seems you made some typos in your reasoning). It should be when the distance between 2 centers is less than or equal 2R, the two circles may still be separated because the set of covered cells is smaller than the original circle.
eh, you are right. I mistyped.. Thank you, already edited.
I also doubt it.
This is a simple example:
R = 2
Center 1 at (3, 3) and center 2 at (6, 5). Distance is sqrt(3 * 3 + 2 * 2) = 3.6 < 4 = 2R
We are investigating the situation.