This challenge, as described from some of the linked pages, is exactly a challenge on algorithms and data structures not on gluing APIs together. The problem with that is that it's exactly the kind of challenge that LLMs do well at because there's a glut of material to train on.

The challenge is figuring out how to provide a novice (high school students) with a skill appropriate challenge to assess them on that is also not trivially solved by the use of an LLM, and you likely can't. Or you go the IT route and restrict what's available on and from the computers (see my comment about how ACM and IEEE contests used to do things, they may still run that way but I'm uninvolved so don't know).

That is a thing in the ACM-ICPC. Typically competitors would come in with printouts of times they've implemented a tricky algorithm like the Hungarian Algorithm or the Blossom Algorithm in the past, so that having it would help jog their memory and let them be constrained only by typing speed once they've figured out how to adapt it to their problem.

It was referred to as "book code."