SIMO Retiree Blog

 (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

(David here.) Continued from Day 1 here. Problem 4 (IMO 2024/Q4) Let $ABC$ be a triangle with $AB < AC < BC$. Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively. Prove that $\angle KIL + \angle YPX = 180^{\circ}$. We spent 25 min drawing diagram in Geogebra (heh). Diagram stolen and modified from Evan Chen's AoPS post. 1. The tangents through $X$ and $Y$ actually concur on $AI$. This is because with $AB$ and $AC$, they form a rhombus. 2. The obvious thing to do to eliminate the two midpoints is to homothety $I$ out 2x to $J$. This gives $\angl

(David here.) Recently, Sheldon and I attempted the IMO questions (with AYS and Aloysius making guest appearances along the way). I've tried to document some thought processes and some mishappenings - compared to the cleaned up solutions you'd see on the AoPS thread, this would be instead a messier look at how one might go about the problems.

(Wee Kean here.) I recently learned some interesting game theory which may or may not (read: will not) be useful in your life. For starters, we begin with the classic game of Nim.

 (Dylan here.) With the IMO drawing near, I thought I'd stray from maths to do some rambling.  What is the IMO? What is tested at the IMO? Do I know enough maths? Why do people care about the IMO? Should I care about the IMO? How do I get to/train for the IMO? How do I know if I'm improving? What happens after the IMO?