I have been working on PE problems for most of 10 years. One thing I would sort of like to do is make a library (mostly I have been using python) for some of the more common functions. Do you have anything like that?
I do, but there's nothing too obscure in it. Efficient prime number sieve, prime factorization using trial division, generating list of divisors from the prime factorization, modular inverse via Euclidean algorithm, Chinese Remainder Theorem.
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.
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.