Back in early October I was already looking forward to the tournament, and on 12th October I reached the official site for the problems.
But no, nothing was on the site.
I waited and waited. Last year I posted my analysis on 20th October assuming that the problem came up a few days prior. I checked the official site and there is no updates other than saying that "it will be held on Oct 12", not even an update that concludes the event. I checked AoPS, no one posted there as well. The social media has been dead since 2023.
It took me a whole month before I finally get my hands on the problems and not even a full set. Someone shared paper A and B assuming the participant was in the east, but I don't see paper C around yet. How hard is it to update your site and release the problems to the public? That's just another disappointment along the years.
I am already travelling partially for the sake of baseball. That's story for another day for sure, but let us focus on these problems first. Since I don't have the luxury spending big chunk of time to solve these problems, my comments tend to be shorter this time.
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A1. I think I have accepted Q1 of this difficulty somehow. WLOG assume $a\geq b \geq c$, then $[a] \geq [b] \geq [c]$. Write $[a] = [c] + k$ for some integer $k$ then you find that $[c](a-[a]) \geq [c]([b]-[c])$ which gives $[b] = [c]$. Similarly $[a] = [b]$ which obviously $a=b=c$. AoPS provided another neat solution too: notice that $\frac{a}{[b]} = \frac{c}{[c]} \geq 1$ giving $[a] \geq [b]$, but circular argument gives $[a] = [b] = [c]$.
A2. The general solution is less obvious than some of the previous Q2s, but the parameter of 2024 is so low that allowed some kind of exhaustion if candidates wish to spend 2 hours on this question. This is a replicate of JBMO2022 Q4 and if you don't want spoilers -- think about the ternary representation.
A3. err...standard analysis assignment? Think about how $\frac{1+na}{1+nb} \to \frac{a}{b}$ which gives you an explicit formula for $m(a,b)$. Once you have that, you can lower bound $n$ in terms of $W$ and $m(a,b)$.
But more importantly, you can do everything without using any abstract idea because intuitive simple parameter testing would give a clear answer as well. No trap and straightforward. Is this really Q3?
A4. Oh...finally a number theory question, but it is one that is insta-killable if you know something or immediately a big fat zero if you don't. This is also why competitions like Putnam doesn't make these questions often -- it simply is a skill check more than anything. And for that reason, I think it is quite dumb to have it here.
Prime numbers...what could be the theory behind other than the prime number theorem? Note that you want a lower bound of density (not the average density), but this is 'easy' if you have seen that before.
That says, you can expect the chance to be lower bounded by $O(1/ \log s_d)$, and since $s_d \leq d^2$ you know the summation diverges. The fact that you need an accurate bound like $O(1/\log p)$ instead of Bertland's postulate shows that A4 really is a dumb question checking your knowledge on PNT.
Please don't do it again.
B1. Typical 'guessing game'. Partition into 12, 34 and 56 and you test 12 and 34. If it reveals 00/02/20 this is over. Otherwise you know a coin is in each of two of the pairs. Then you test a single box by choosing an empty box from the empty pair.
e.g. I know a coin is in 12 and a coin is in 34. On the third guess I would try 15 -- if it gives 1 coin then I know box 1 contains a coin, if it gives 0 coins then I know box 2 contains a coin. Then I try 35 for the same reason.
And why is 4 guesses the minimal? Well there are 15 possibilities. Although in theory 3 guesses gives 3^3 outcomes there aren't that many actually because once you get "2 coins" the guess is over, so it does not distinguish as much.
B2. Try the inverval $x\in (0,1)$ first. By continuity you will notice that they are all determined by the value of $f(0)$ - a simple, single parameter function. Keep on doing that for $x \in (-1,0)$, then $f(-1)$ and $f(0)$, then $(-\infty, -1)$ and $(1,\infty)$. Again routine.
B3. Duality and projective space! This is on the borderline of skill check like A4 because the construction if there is one, should not be too exotic.
We know that there is a canonical bijection between lines and points in RP2, but Euclidean planes do not contain points/lines at infinity that the duality fails. However this question isn't really asking for duality -- we only need a correspondence that works for distinct non-parallel lines, and that's the technical difference to take care of.
B4. Oooof, such a nicely formulated problem surrounding symmetric polynomials yet so horrifying to look at. Not going to try this.
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Problems appears to be more polarized: some are clearly high school (if not junior) level MO problems, but some are extremely specific problems at senior undergrad level. Is that somewhat expected? Probably. Does that make the tournament more Putnam like? Hell no.
Guess I will write more about problems this year when I have my hands on paper C, or when they finally decided to release the problems and answers officially. See you ^.<
