SIMO Retiree Blog

  It's Glen again. At last, I have time to write this out. Again two of these were done without paper but I've tried to reconstruct the thought process as faithfully as possible. The wording might be a little fuzzy because I'm transcribing pictures in my head. Probably everything would have been better written out if I had given names to everything. Apologies in advance. Problem 4 ( IMO 2026/4 ) Shan-Yu and Mulan are playing a game. Let $\theta$ be an angle with $0^\circ<\theta<180^\circ$ known to both players. Initially, Shan-Yu makes a paper triangle $\mathcal{T}$ with measurements of his choice. Then, they repeatedly perform the following steps: If $\mathcal{T}$ has at least one angle measuring exactly $\theta$, then the game stops and Mulan wins. Otherwise, Mulan chooses a point $P$ on the perimeter of $\mathcal{T}$, different from its three vertices. She then makes a straight cut from $P$ to the opposite vertex of $\mathcal{T}$, splitting it into two triangles. S...

  Glen here. Unfortunately due to other commitments, I was unable to be at IMO this year, but I did have time today to attempt the problems from Day 1. I'll be writing about my thought process in solving these problems, but I did two of these (the non-geometries) in my head so certain ideas might have been forgotten. Also, I might be a little terse because I want to go to sleep. Problem 1 ( IMO 2026/1 ) There are $2026$ integers greater than $1$ written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m>1$ and $n>1$ from different places on the blackboard and replaces these two integers with $$\gcd(m,n) \quad \text{ and } \quad  \frac{\mathrm{lcm}(m,n)}{\gcd(m,n)}.$$ He continues to make moves while it is possible to do so. (a) Prove that, regardless of the choices of Confucius, after finitely many moves, exactly one integer $M$ on the blackboard is greater than $1$. (b) Prove that the value of $M$ does not depend on the choices of C...

Glen & Sheldon here. At last, our schedules aligned enough for us to try this year's SMO Open together. Here's an outline of our thought process while solving the problems, featuring some screenshots from our Cocreate whiteboard. Problem 1 In the triangle $ABC$, $\angle B>90^\circ$, the incircle touches the sides $BC$ and $CA$ at $D$ and $E$, respectively. The lines $ED$ and $AB$ intersect at $P$. The incircle of the triangle $AEP$ touches the sides $PE$ and $AP$ at $D_1$ and $E_1$, respectively. The lines $E_1D_1$ and $AE$ intersect at $P_1$. Suppose $P,C,E,B$ are concyclic. Prove that $BE$ is parallel to $PP_1$. Weird-looking problem. The concyclic condition means $\angle ABC = \angle AED$. Maybe it could be less weird if we write it in terms of the angles of the triangle, $\angle ABC = 90^\circ + \frac{\angle ACB}2$. Actually, the equal angles imply $\triangle ABC \sim \triangle AEP$. The parallel condition is equivalent to $\frac{AB}{AE} = \frac{AP}{AP_1}$, and the s...

(Drew here.) As a retired contestant, I decided it would be fun to attempt the IMO 2025 paper and see how well I would do on it! Let's get started. Problem 1 (IMO 2025/1) A line in the plane is called sunny if it is not parallel to any of the $x$–axis, the $y$–axis, or the line $x+y=0$.  Let $n\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following: for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and exactly $k$ of the $n$ lines are sunny. To begin, I decided to try the $n=3$ case, as that's the smallest one. The points form a right triangle with $3$ points on each edge, and a sunny line is a line not parallel to any of the edges of the main triangle. We can shift the points a bit to instead form an equilateral triangle, and a sunny line is any line not parallel to any of the sides of the equilateral triangle. Important...