SIMO Retiree Blog

(Drew here.) As a retired contestant, I decided it would be fun to attempt the IMO 2025 paper and see how well I would do on it! Let's get started. Problem 1 (IMO 2025/1) A line in the plane is called sunny if it is not parallel to any of the $x$–axis, the $y$–axis, or the line $x+y=0$.  Let $n\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following: for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and exactly $k$ of the $n$ lines are sunny. To begin, I decided to try the $n=3$ case, as that's the smallest one. The points form a right triangle with $3$ points on each edge, and a sunny line is a line not parallel to any of the edges of the main triangle. We can shift the points a bit to instead form an equilateral triangle, and a sunny line is any line not parallel to any of the sides of the equilateral triangle. Important...

(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.

(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.

 (This is Glen.) At some point, Ker Yang wrote a post on Reim's theorem, which he used to solve two past year IMO problems. I remember commenting to him that I had solved neither of them with Reim (and no, I did not bash them). Later on, I tried reconstructing my solution to IMO 2017/4, and I noticed something interesting that made me find another (slightly weird) solution that (I think) isn't on AoPS. So that's what I'll be writing about today.

Etienne here. Today I want to talk about Problem 6 of the IMO, and my thought process while solving it. The official IMO document provides 7 solutions to this problem, and to me all of them feel strange and unmotivatable at first read (especially the construction!). So I hope to demystify this problem and present it in the simplest and most intuitive way possible. The Problem P6. Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$, $$ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). $$ Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$, and find the smallest possible value of $c$. Solution This problem statement should jump out to anyone as unusual. First, for each pair $(x,y)$, $f$ can choose between two equat...

(This week: a special guest post from Andrew, a member of the IMO 2024 PSC!) This post is going to be a discussion of everything that could possibly be construed as related to Turbo , i.e. IMO 2024/5, from my perspective. It will be long. Apparently I will also be up against basically all of the Singaporean old people, so this should be fun. (Spoilers for IMO 2024/5. The opinions represented here are those of an individual and not necessarily the opinion shared by all or even a majority of PSC members.)  Turbo (IMO 2024/5) Turbo the snail plays a game on a board with $2024$ rows and $2023$ columns. There are hidden monsters in $2022$ of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster. Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in t...

 (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...