SIMO Retiree Blog

(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...

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...