SIMO Retiree Blog

Glen here. At some point about a year ago, I scrolled through the blogger interface and found an old draft by Dylan, a skeletal outline of a post featuring a problem and a few section headings. The problem happened to be one I recognised: Show that $2^{2000}$ divides the numerator of $2+\frac{2^2}2 + \frac{2^3}3 + \cdots + \frac{2^{2024}}{2024}$ when expressed as a simplified improper fraction. I remembered first seeing this as a bonus problem in some old SIMO National Team set, when I'd looked at it, had absolutely no idea how to approach it, and promptly given up. But this time, having learnt some more math in the intervening years, I did  have an idea of what to do, and ended up solving it it in my head during some seminar that I wasn't quite following. The solution turned out to be pretty cool, so I made a note to myself to write a blog post about it at some point if Dylan's post never manifested. I then promptly forgot about it. Fast forward to this week. Some of the i...

(This is Glen.) I was sorting through the LaTeX files on my computer and unearthed an old set of solutions to a mysterious problem set, dated 2020. After a bit of digging in some old Discord servers, I found out that these were solutions to one of Zhao Yu's sets for some RI training, which I had presumably crashed because it was online (thanks to Covid) and I was too free or something. Anyway, this file contained a lengthy introduction to UFDs, which I had recently learnt about in uni and had used to overkill a couple of problems in the set. This is, I think, quite suitable for a blog post, so here we are. The fundamental theorem of arithmetic As a warmup, let's think about something we learn about in primary school (well, at least I remember learning about this in primary school but I am old so this may no longer be the case): the unique prime factorisation of integers. (Fundamental theorem of arithmetic) Each integer $n>1$ can be written uniquely as $n=p_1\cdots p_k$, wher...

 (Zhao Yu here). Today someone spoke about this short proof of a lower bound of the number of primes, which I found it too nice to not share.

(Drew here.) I've spent a lot of time browsing random math pages on Wikipedia  (blue link). Sometimes I find things which are actually useful in olympiads, but a lot of the time there's this void of open problems. Of these open problems, some were only solved recently, and they can be used to blast olympiad problems.

Hi, Choo Ray here. Before IMO 2024, the Singapore team had a joint friendly contest with Iran and Taiwan (spoilers ahead). As I found out, it takes quite a lot of work to run a contest. We wanted to simulate the IMO format in terms of topic diversity and difficulty but our shortlist lacked depth in certain topics.

(Jit here.) I am going to write about some recent ideas in diophantine geometry. Consider a curve $C$ inside $\mathbb{C}^2$ given by some polynomial $P(x,y) = 0$. Can there be infinitely many points $(x,y)$ lying on our curve $C$ such that both $x$ and $y$ are roots of unity?

(David here.) In this post I want to share a really interesting factoring algorithm that I learnt while taking an advanced cryptography class , and it will serve as a good springboard into talking about the mathematics and algorithms of lattices.

 (Jit here). There is a famous theorem of Lagrange that says every natural number $n$ is a sum of four non-negative squares (so that $0$ is allowed). Let's try to prove this theorem.

(Zhao Yu here.) It's been a long time since I've touched olympiads and I have a really bad memory, so today I'll try something different and talk about something unrelated to olympiads. I will talk about 3 interesting facts that follow from the theory of elliptic curves . I will not be precise, and I will black-box many things. The point is just to give you a taste of elliptic curves and hopefully this will motivate you to learn the actual theory in the future!

 (Dylan here.) For this post (and perhaps my subsequent blog posts), I will share a math-adjacent slice of my experience, and then talk some (probably unrelated) math. I will also include upfront some problems related to the non-math/math I am talking about, in case you want to try them first.

(Jit here). Today I am going to write about the Pell's equation $x^2 - dy^2 = 1$, which hopefully should be familiar to readers, and about generalizations of it to higher degrees. Let's first review some facts about the integer solutions of a Pell's equation.

Wee Kean here. Happy New Years Eve! Lately, my only interaction with math has been through doing Project Euler. So... here's some cool things I've learnt! Unfortunately, there is hardly any structure to this blogpost so I apologize if this comes off as verbal diarrhea.

Hey! Choo Ray here. I've been involved in giving sessions for the SIMO National Team and other trainings recently, so naturally I have to hunt for suitable problems. I feel that I've been looking at more problems recently than I have in the lead-up to my participations in IMO/other competitions! Of course, I spend less time on each problem, as my objectives are geared towards discovering problems with good ideas and instructive value rather than solving them myself. However, sometimes I find myself being led down a long rabbit-hole of theory that I apparently ought to know about. In this post I'd like to share about one of these experiences. One day, I was browsing contest collections on Art Of Problem Solving (AoPS). A question from the 2020 Bulgarian National Olympiad caught my eye: P4. Are there positive integers $m>4$ and $n$, such that a) ${m \choose 3}=n^2$ b) ${m \choose 4}=n^2+9$ I clicked on the link, thinking that it seems a rather routine proble...

(Jit here). Today I want to explain how to prove Skolem-Mahler-Lech using the p-adic numbers, and explain some sort of generalization in the area of algebraic/arithmetic dynamics, known as the Dynamical Mordell-Lang conjecture, which is still an open problem.

(David here!) I wanted to talk about a problem I encountered in 2020 during Putnam seminar, and the subsequent exploration I did on it. It's still unresolved but I hope my thought processes could help those who are interested in understanding how to propose problems.