I must be dumb, because, after reading the problem statement, I'm not sure how it's possible that even 1 (much less 99) of the 100 mathematicians are going to be able to guess their unopened box correctly.

All 100 mathematicians are able to confer with one another before separating, but once alone, each person is just as clueless to the contents of the boxes as any other person, and they can't communicate with one another.  They could all open all of the boxes in their respective rooms, save one.  But of course, the rationals are no-where dense in the reals, if I recall correctly, so there's still probability zero that they'd guess the contents of the last remaining box.

>> Can anyone enlighten me?

Clearly not me.  But Michael may be able to.

Here is a general thought...

When you get something like this that looks utterly impossible or at the very least highly counter-intuitive, here is what I'd suggest.

Try and prove it's impossible. Precisely and rigorously, and with no hand-waving.

If you fail, look at where your best attempt at a proof breaks down and try and use this to construct a solution. If you do think you've managed to prove it's impossible, scrutinize your proof very carefully for the tiniest gap. If you find one, try and use that loophole to make a solution (see above). If you can't find a gap, maybe you've proven it really is impossible.