I don't know if I'm that smart... That test was special, because there was really nothing to calculate on most of the exercises. That test was actually pretty ideal for me, because I'm already a 2nd year graduate student, though in Europe. I could answer about half of the questions directly without thinking about them at all. If you look at the questions, all the theoretical claims can be disproved or proved by taking pretty much the canonical example of the structure in question.

About my background, I have a M.Sc. in math, which is a 5 year degree, but I finished it in less than 3 years. My GPA is 5/5 in maths and 4.94/5 overall. I've already taken quite a lot of graduate math, e.g. commutative algebra, homological algebra, fields and galois theory, measure theory, functional analysis, algebraic geometry (schemes).

I studied for the GRE by buying the Princeton Review book two weeks ago. I had planned to study for about a month, but there were some personal things that got in the way.

My weakness is that I haven't done any practical calculations since the second year of my undergraduate, so I had forgot most of the formulas. Formulas for e.g. Fourier series and how they work I nowadays only remember by thinking about how it relates to the projection to a subset with an orthonormal basis and then look up formulas from a book. Similarly sums of trigonometric formulas I relate to Euler's identity. However, the GRE doesn't care about the fact that you know how to derive everything, you just need to know the formulas.