Impossible Masses

Man, two weeks without posting, who's driving this thing? Anyway, time for a new puzzle. I first learned this one from the math room on KGS:
Consider you have a collection of 11 weights, each with a rational mass. This collection has the property that if you remove any one weight the remaining 10 can be sorted into two piles of 5 weights each, and each with the same total mass. Prove that this is impossible unless every one of the weights has the same mass.

By "a rational mass" I really mean that the ratio of any two of their masses is rational. It feels a bit weird to ask for a proof in a logic puzzle, but the proof is really quite elegant, and understandable without complicated math.

It can also be proven when the masses are real instead of rational, but it is much more abstract to do that.

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