Heh, as someone who's working behind the scenes, it's nice to see HMMT problems being discussed here.

Here's one of my favorite math competition problems of all time (bonus points if you can find the source):

We have a sequence of real numbers a(0), a(1), a(2), ... that satisfies a(n) = r * a(n-1) + s * a(n-2) for some real numbers r,s. Is it possible that a(n) is never 0, but for any positive real number x we can find integers i and j such that |a(i)|< x < |a(j)| ?