5M. This problem is quite difficult for J1, but I think a lot of the perceived difficulty of this problem (many people rating this problem 10M or even 15M) comes from its placement and the psychology of "this problem should be easy" and getting tunnel vision. There are a few ways to solve this problem, and none of them are too hard to see or too complicated, so I don't think this deserves a rating of 10M or higher. It is a really weird problem though; it's not straightforward like most J1s are, and it's definitely harder than most problems of its position.

10M. Many people solved this problem quickly (often quicker than J1), so a lot of people rated this 5M. However, I don't think this problem warrants a rating of 5M because it's a very counterintuitive problem (when I read this problem, I stared at the problem statement for a good 5 minutes think "wait, this is true?") and it takes a bit of time to really understand what's going on in the problem. I also think it's really easy to get sidetracked in this problem - I spent about 20 minutes on this problem trying to formalize the division algorithm argument before seeing a much easier approach. That said, it's not a very difficult problem, and fairly typical of it's placement.

25M. This problem is easily the hardest on the test, though seeing how I managed to solve it, it's not incredibly hard. This problem essentially has two parts - proving that the checkerboard coloring is necessary and sufficient to determine domino-tileability and finding the number of paths that satisfy this checkerboard coloring (although some other, less direct approaches are different in this nature). Both parts are nontrivial and solving one part doesn't necessarily give insight into the other (again, other approaches are different, though I think these are harder to find).

5M. Like J1, this is quite difficult for J4, though unlike J1, many people solved this quickly. The induction solution is very straightforward, and looking back, my initial rating of 15M was mostly because I didn't find it (and that I'm just terrible at inequalities in general). My solution, being the notorious combo main that I am, was a combinatorial argument on coins and tokens, which I found after 30 minutes of bashing out algebra just to find that my approach didn't work :clown:. However, the induction solution is still not exactly trivial to notice, so I think a rating of 0M is inappropriate for this problem. (I actually was debating between rating this 5M and 10M before settling on 5M.)

5M. Given that I actually managed to solve this fairly quickly, this geo is not difficult at all. Configuration issues are not a problem and there are many different fairly easy synthetic solutions as well as ways to complex and coordbash. However, I am rating this 5M rather than 0M because some solutions rely on the well-known fact that P lies on (ABC) and all solutions require some additional insight beyond just looking at the diagram without constructing any new points, as is typical for a 0M geo.

20M. This problem looks pretty easy, but it has annoying edge cases in at least two different solutions. I rate this 20M because the edge cases are really annoying and have to be dealt with individually, and they are not hard to miss. It's fairly easy to get partials on this problem due to the main difficulty being, in my opinion, the edge cases, but getting the full 7 points is difficult and I think this warrants a rating of 20M.