A man offered me a set of eleven weights, not all them equal, each an integer number of pounds, which he said had the following property: if you removed any of the eleven weights, the other ten could form two five weights sets that balanced each other. Is this possible?

And if the weights didn't weigh an integer number of pounds each?

One part 1, both McWorter's proof and e.g.'s infinite descent proof
work for me. I have a preference for McWorter's proof, since it
does not involve 0 weights, which are more theoretical than I care
for. Well done!

On part 2, I am sorry this thread died. I still don't know
whether this is possible or not, but I agree with Tristan that it is
unlikely, since it would need to involve irrational weights.
(Otherwise, we could multiply to get an integer solution). I
haven't figured out a solution, but I haven't figured out an
impossibility proof, either.

So, Great problem, Oskar!

Do we get a hint?