SIMO Retiree Blog

(David here.) Recently, a problem of mine came out as Q4 on SMO Open 2024. I'll go over how I proposed the problem and the theory behind it, which involves some machine learning ideas (!).

(Pengchong here). While in NS, I've been looking at some topics that I might've encountered back in school but not really gone too deeply into - one of which is real analysis, quite a common starting point for undergrad math. Though for those preparing for olympiads it probably doesn't have a high 'ROI', I think much of this stuff provides plenty of intuition for some olympiad problems I've encountered before. And of course, most importantly, I found it pretty fun (looking at Rudin's classic book, in particular). I'll give a brief introduction to some fundamental analysis concepts using ISL 2020 A8 as the vessel, and as an example of how these concepts can sometimes (though rarely) turn out pretty critical for a 'real' olympiad problem. I've used this problem during a session for this year's IMO team a while back (and actually got this in a mock during my own IMO training). It will only use the very fundamen...

 (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, es...

(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 ...

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...