Ok, so this problem looks like an induction problem to me, so I used that, but I only got as far as the induction hypothesis. The hint says to use the pigeon hole principle. I'm not sure how to use that for this problem.
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I don't know the solution to this problem, but here are a few things that might be useful:
There are only 2m distinct absolute values available between -(2m-1) and +(2m-1). Therefore if you have 2m+1 distinct numbers, then an absolute value must be repeated. This means that there must be a positive and negative pair. In other words, the set must contain two distinct values j and -j. If the set also contains zero, then you're done.
Therefore, we may assume that zero is not one of the 2m+1 distinct integers. This means that there are in fact only 2m-1 possible absolute values (since zero is excluded). Therefore there must be two repeated absolute values, i.e. two positive/negative pairs.
I haven't worked out where to go from here, and it might be a dead end. It might be useful to play around with a small concrete case, say m=4 or 5, and see how you might prove it in that case.