SIMO Retiree Blog

This is Glen again. There's been a lack of regular olympiad content recently, and since no one posted this week I thought I'd attempt a problem and write about my thought process. There was some discussion recently on the retirees' Discord server about APMO, so I figured I'd look at the most recent iteration, and since I wanted to be able to solve the problem in a reasonable amount of time, I attempted Q1. Here's the problem: ( APMO 2024/1 ) Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear. Preliminary thoughts First and foremost, here's a diagram: Sorry about th...

(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 ...