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If your example is only vectors (0 , 1) and (1000 , -1000) - the answer is only vector (1000 , -1000). Give me some test for which this idea is not correct.

What about this...
For some vector V you know until which vector U it grows (until degree between V and U is <= 90!?). Using Binary Search you can find U. Before looking for result you can find sum of first i vectors i = 1 .. N and in O(1) for each pair of vectors you can find max distance. It's n ...

Maybe this can help you
http://forums.topcoder.com/?module=Thre ... 123#534442