IMO 2026 Day 1 Livesolve
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...