@BonusQuestion – well, thinking it through we can assume that if we can’t find a set that some part doesn’t add up to zero or 2009 in a set where the integers all have an absolute value of 2009 or less individually, then it really doesn’t matter how much larger you make the numbers, you’re essentially just increasing (or decreasing) by some multiple of 2009, correct?
So, put another way, whether I use 3 or 2012, or 4021 really doesn’t matter since they would all have an “actual impact” (my own made up technical term) of 3… the same with negative integers…
So we can’t use two numbers with the same absolute value, since we automatically create a subset that adds up to zero, so our dream set would essentially be something that alternates the sign while increasing the integer (1, -2, 3, -4….) but since the set is so large you end up with the last few values being (... -2006, 2007, -2008, 2009) so we have to directly include 2009 in the set, if we push the set one down the number line, we have to include zero. If we just jump over zero and 2009, then we include 2010 add 1 and -2 to get 2009 again. If we add -2010 instead then we can just add 1 and get -2009.
How to write that up in mathematical notation is beyond me.