SIMO Retiree Blog

It's Glen again. Last week , I mentioned how the relative difficulty levels of the ISL 2018 C4, C5 and C7 felt off to me. Out of curiosity, I looked up the C6 to see where it'd fit, and subsequently ended up solving it in the shower. (Yes, recurring theme, I know.) Anyway, this post will be about my thought process in solving this problem (sans paper, because shower). ( ISL 2018 C6 ) Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. If no such pair exists, we write two times the number $0$. Prove that, no matter how we make the choices in (i), operation (ii) will be performed only finitely many times. Preliminary thoughts Here's a way of visualising this problem: we can think of representing our numbers as chips on a number line. Then, operation (i) corresponds to...

(This is Glen.) I was sorting through my stuff and found a piece of paper from when I tried last year's NTST problems, which I presumably saved because I thought I might eventually want to write a post about it. So here I am, writing a post so I can safely dump this piece of paper into my recycling bin.

(Dylan here.) I realised the general content on this blog is pitched at a rather high level. Here's an attempt to balance it.

 (Drew here.) Three weeks ago, we had a post about visualising combi, and last week, it was about the anatomy of an FE. Out of sheer coincidence, here I am now talking about visualising FEs! A nice way to avoid the notoriously grindy nature of FEs and favour a combi main like me :)

 (Glen here.) I had the privilege of being a Coordinator at this year's IMO, which, along with some extra perks such as a lot of free food and being able to witness Singapore's 3rd-best ever team placement (!), came with the benefit of seeing this year's IMO problems a couple of days before everyone else. This also meant that I had a couple of days to make predictions about how each problem would be received, and I turned out to be wildly wrong about one problem in particular: Problem 3. I'd found this a lot easier than its single-digit solve-rate would suggest (this took me less than an hour without paper), and so I figured that I should write something about my thought process behind this problem. For obvious reasons, I don't have an actual record of solving the problem, but everything has been reconstructed to the best of my memory. The problem I first saw this problem on the Saturday before the IMO. There was a nice fancy dinner for Coordinators and Leaders, and...

(Xu Chen here) Today I am going to talk about a subtle but important thought process for many problems, which is how you visualize the objects you are working with. This is more of an intuitive idea than rigorous math, so forgive me if not everything is super clear in this post. How you look at the problem can heavily influence what ideas you can come up with: The solution often follows from a good understanding of the underlying structure. Example 1. Let's start with a simple example. (USAJMO 2019/2) Let $\Z$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f:\Z\to\Z$ and $g:\Z\to\Z$ satisfying $$f(g(x))=x+a\quad \text{and}\quad g(f(x))=x+b$$ for all integers $x$. Hopefully it is clear that we cannot treat this as a standard FE, but must instead consider some combinatorial ideas. Let's think of $\Z$ on the number line. We draw red arrows to represent $x\mapsto f(x)$ and blue arrows to represent $x\mapsto g(x)$: One cond...