What was your strategy like? How much math background do you have?
I attacked roughly the first 250 problems in order. The early problems build on each other to introduce new topics. I also got good at figuring out the right search term to find some random paper in number theory, combinatorics, probability, whatever.
Later problems introduced new, more niche areas, like chromatic polynomials and impartial & partisan game theory. But by then, I found it much easier to figure out what part of math a problem was based on and how to find relevant literature.
It helps to be really really stubborn, and to have the patience to let a problem stew in my brain, sometimes for weeks at a time. That seems to help lead to that Eureka moment.
As for a single problem, I'm fond of PE589, "Poohsticks Marathon". That was my 501st solution, two years after first attempting it (solved 5 years ago, yikes). I like it because it's a problem with a 95% difficulty rating, so very tough, but the development team slotted it in as an easy problem (problems normally get scheduled in batches of 6 with a cadence of medium/easy/medium/easy/medium/hard). Once I solved it, I agreed that it was relatively easy, in that it uses techniques introduced by early PE problems, but something about it makes using those techniques unexpectedly difficult.
I also have some stand-alone modules, one to solve generalized Pell equations, another to find a polynomial given a sequence via the differences (e.g. 2, 5, 10, 17, first differences 3, 5, 7, second 2, 2 is enough to find n^2+1). There's another to find the closed form for a sequence as a linear recurrence.
Some solvers have much more extensive libraries, but I tend to grab bits of code from old solutions to reuse on the fly.