I got silver when I solved my first one (edit: I got into 'first 250' then).
So maybe it's like this:
Bronze - solve at least one
Silver - be in the 'first 250' at least once
Gold - top 100 OR 'first 250' three times
Then again I never saw anyone with a bronze one, so that's only a hypothesis
You're welcome, btw. Wish we could reveal our real selves, but Neopets are probably monitoring this topic