SIMO Retiree Blog

(This is Glen.) I'm doing this since I'm relatively free at the moment (it's still summer vacation) and I've been looking for an excuse to try these problems anyway. I'll give myself 4 hours (which I think is the duration of the competition?), type everything I'm thinking of in bullet points and add headers to make things more readable at the end. I'm not writing out solutions properly because the focus is on the thought process rather than the actual solution but the intent is that the solution is still discernable if you follow everything I write down. All in all, the time saved in not having to write things nicely should cancel out with the extra time I spend typing out all of my thoughts. Anyway, here I go. Let's hope I don't embarrass myself. Problem 1 In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$; lines $AB$ and $CM$ intersect at $P$; and lines $AC$ and $BM$ i

(Dylan here.) Today, I'll be solving a (pseudorandomly chosen) combinatorics problem. But instead of just explaining the solution, I will detail my messy thought process as I work through it in real time.  After solving the problem, I will do a brief meta-commentary, and try to abstract the problem. Finally, for the more mathematically mature, I will talk about what possibly lies beyond it: various extensions and related areas. Let's begin. 1     The Problem (APMO 2023 P1) Let $n\geq 5$ be an integer. Consider $n$ squares with side lengths $1,2,\dots,n$ respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly $2$ other squares.  1.1     First Thoughts In no particular order: This is a  construction  problem.  The orientation of the squares are fixed to be aligned; the

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