interesting math problem

[gumpzilla]

I realize that these math problems aren't everybody's cup of tea, but I think there are a few OOTiots who might enjoy this one. I grabbed it from the USSR Olympiad Problem book.

Suppose you have thirteen gears, each of which has integer weight. You know that these gears have the property that if you choose any twelve, you can separate that group of twelve into two groups of six which will be of equal weight. Prove that all of the gears must have the same weight.

Feel free to post reasoning as to how to go about this, and if you have seen it before, hold back for a bit, I guess. (I haven't looked up the solution yet, nor fleshed out my argument totally rigorously, but I'm pretty sure that I now understand one mechanism for proving this)