## Thursday, 17 November 2022

### Simon Marais 2022

So this year I am not almost a year late. It's always a joy for me to deal with these questions. Here are my immediate thoughts after reading the scripts -- immediate means that I didn't put any serious effort into it and didn't do any calculations that requires pens and papers.

One changes they made was to split the tournament into two time zones to reduce the chance of possible cheating but I can only say that it's far to easy to cheat at institutional level if they wanted to...but the good news is that we will get four more questions to solve.

A2. A very nice combinatorial little question but should be A1 in my eyes. The $e^{-2}$ made me think of derangement since $n!/!n \approx e$ but no luck in that. Eventually you realize that the question is much much easier -- this is just the chance of an item appearing in at least one but not all $n$ subsets, for all $2^n$ elements.

A3. IVT because why not? But I don't like this question due to the lack of elegance...

A4. Two parts in the question. Showing that to be an integer is a standard high school MO question. The second part...something with field if it's put as a Q4? Bad luck if you didn't do enough math olympiad in high school.

B2. Question that I love so much because imagine getting that in the 1985 A-level (HKALE) pure maths paper! This is not a joke and is totally possible. Since the non-zero terms are of even degree, the roots are symmetrical along the real axis, hence the circle centre is real. That means the roots are on a circle centered at origin upon a real translation. Boom!

B3. Easy if you consider the "1D" version first. This game can't be any simpler...

B4. (a) is easy as a Q4a (unless I made a mistake): classify $2[b_n/20]-3[a_n/30]$ based on $a_n-b_n$ in intervals of 10. Since the difference between that term and $(b_n-a_n)/10$ is at most 3 that gives you the idea of splitting the cases $a_n-b_n$ being either between -10 and 20, above 20 or below -10. The first case gives the induction and the latter cases gives strict monotonicity which gives the claim.

(b) is surprisingly elegant as an open question. It is not easy to think of other open problems as simple as this other than the 3n+1 conjecture.

C2. ...why? Instead of a A-level pure maths question you give a calculus I assignment question... The ceiling function means that ceil(sin x) is either 0 or 1. The irrationality of $\pi$ means that the term is negative, hence rounded up to 0, at least $0.5 - \varepsilon$ of the time. If irrationality is too hard, it is still easy to argue that at least one out of four consecutive terms is of power -1 (diverges) instead of -2 (converges), then the comparison with $\sum \frac{1}{4n}$ gives the desired result.

C3. Classic MO question again. If you are not sure, consult lower cases like $n = 7, 12, 17...$ etc.

C4. Another good question but looks quite intimating other than the $(n-1)!$ term who's there for the obvious reason.

You ask me why aren't there Q1s above? The answer is clear: they do not worth talking at all. Q2s are very easy (probably of Q1 level in past papers or even lower) in my eyes but Q1 are even easier. They looked ugly and tedious, but not hard. Q3s are easy to moderate, but I don't think they are as hard as Q3s in the past. Q4s are hard as they should be, but that means the Q3-Q4 gap is even larger.

This is not a very good contest paper not just in terms of difficulty spread, but also in terms of the scope -- many of the questions are plain high school MO level. Take 2021 putnam for example and check what they used: non-integral binomial theorem, qZ rings, double integrals, fancy polynomial about primality, geometric probability, infinite sum, Green's theorem (if you don't recognize B3), matrices and determinants. These are things that contestants are expected to use inside that examination. They should really expand in terms of number of questions with more challenging ones (but not as hard as those Q4s) and make sure that college maths were cooperated into the questions...