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...