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To Spoil or Not to Spoil (ft. A7, the most beautiful problem of ISL '23)

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 (Dylan here.) As usual, some non-maths, followed by some maths. I. To Spoil or Not to Spoil? Some context, so everyone is on the same page: To spoil a maths problem is to reveal key ideas, steps, and/or details of its solution to others, without their consent.  The content of the problem that is conveyed is called a spoiler (often confused with the thing at the back of race cars that prevent them from flying!). Spoilers could be as short as one word, a picture, or a non-verbal cue. Spoiling could happen anytime there is more than one person in the same room doing maths. It is very common in an unsupervised classroom of high school students. The questions I wish to answer (or at least, bring up for discussion): Do spoilers help or impede learning?  How is spoiling different from teaching? When is is okay to give up on a maths problem? Why do people spoil maths problems? A. Arguments for and against Okay, so no one is actually debating this. Spoiling olympiad problems, especially in a
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Sorting books on a shelf

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(This is Yan Sheng.) The time has come for me to finally write a post about actually solving an olympiad-level problem, in particular the thought process and the winding path that got me to the solution. This came from one of my Discord servers a few months ago: You have a bookshelf of (finitely many) unsorted books. Every minute you are forced to move one book not already in the correct location to its correct (absolute) location, shifting the other books as you do so. This is the only allowed movement. If the books ever become sorted, the world will explode. Will the world explode? (For clarity, all the books have the same thickness, and the bookshelf has exactly enough space for all the books.) Here's an example of the above operation: suppose the bookshelf looks like 312 (i.e., book 3 is in position 1, book 1 in position 2, book 2 in position 3). I can choose to pick up book 2 to move it to position 2, while pushing book 1 to the right, so the bookshelf now looks like
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How I Created a Math Olympiad Problem

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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. Less than a month from the IMO, we were scrambling to have the problems ready. In the end, we cobbled together a paper over some Discord conversations with our friends from Iran and Taiwan. A few months prior, I was jogging at Botanic Gardens, which left my mind free. So I thought, what if we took a positive integer and added its totient function? I was hoping for this to have nice properties since the totient function is closely related to the original number and its prime factors. Inspired by a problem from the National Team Selection Test that had just concluded, I asked myself: does such a sequence always contain infinitely many prime factors? By the end of my jog I ca
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What I could have done better: an X-man's advice to future generations

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(Shi Cheng here.) Hi, it's been a long time since I've been involved in the Maths Olympiad sphere, mostly because of uni, but recently I found out about this blog and I wanted to contribute. This is not a post about Math, but rather, I wanted to share some important mistakes and lessons I learnt from my time as a Math Olympiad student, so that hopefully you will not make them yourself and can maximise your potential as a student. Now, it is possible some of these lessons may already be obvious to you, and perhaps you are judging me for not having realised them earlier. Maybe I should have. But there could be others who may not have realised them, so hopefully this post will stop them from falling into the same traps I did, and for those who are already aware of them, this could serve as a reinforcement. Focus on the big idea(s) of a question. When I was still a Math Olympiad student, I remembered the way I would read solutions is by examining each line, making sure I understo
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Visualising FEs!

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 (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 :) Let's begin with a familiar one: Cauchy's functional equation. (Positive real Cauchy) Find all $f:\mathbb R^+\to\mathbb R^+$ satisfying $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb R^+$. The solutions to this are $f(x)=cx$ for $c\in\mathbb R^+$, which are all linear functions. For any integer $n$, we can calculate $$\begin{align*}f(nx)&=f((n-1)x+x)\\&=f((n-1)x)+f(x)\\&=f((n-2)x)+f(x)+f(x)\\&=\dots\\&=\underbrace{f(x)+f(x)+\dots+f(x)}_n\\&=nf(x).\end{align*}$$ So given a rational number $r=\frac mn$ we have that \[f(rx)=f\left(\frac mnx\right)=mf\left(\frac xn\right)=\frac mnf\left(\frac xn\cdot n\right)=\frac mnf(x)=rf(x).\] Now back to the main topic, let's graph this functi
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Anatomy of an FE

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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
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