SIMO Retiree Blog

(It's Glen again.) Last week, I wrote about my experience test-solving EGMO 2025/3 . As it turned out, I had some free time at EGMO (the night before Day 1, i.e. after we made the mark scheme for Problem 3, and the morning of Day 1 itself) to try the other problems, so I figured I'd write about them as well. I will try to be a little more brief than last week, so the post doesn't turn into a saga. Problem 1 ( EGMO 2025/1 ) For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$ for all $1 \leqslant i \leqslant m-1$ Disclaimer: this was a pretty embarrassing case of wrong reasoning leading to the right answer. Try to spot where I went wrong! (Hint: there are many such places.) Let's try small cases: $N=3,4$ work. $N=5$ fails. $N=6$ works. If $N$ is odd, then the sequence starts with $N=1,2,\ldots$, so $3|N$. $N=3^k$ works: a...

(Glen here.) Last week, I had the privilege of coordinating for the European Girls' Mathematical Olympiad , which meant that I was involved in creating the marking scheme and grading scripts for one of the problems. But before doing any of that, I first had to test-solve the problem I was assigned. This post will be about my thought process during this test-solve. Diagrams are scanned in from my rough work, so I'm sorry if they look horrible. First, the problem: ( EGMO 2025/3 ) Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic. Initial thoughts Here's a d...

(This is David.) I'm back with a short post about a beautiful proof for a beautiful problem I saw recently. Three dimensions? Let me explain the title. I think it was during a decent IMO where Grant Sanderson (of 3blue1brown fame) gave a talk about problems that are super easy once we move to a higher dimension. If you weren't there at the talk, he also made it into a youtube video - I highly recommend watching it if you haven't already! Here at the SIMO X-Men blog, we aren't unfamilliar with this idea - one of the most popular blogposts to date is Glen's Spacetime, Special Relativity, and a Lot of Circles where we saw that interpreting circles as points in 3-dimensional space was a really powerful tool for lots of geometry problems involving tangent circles. And the nice thing is, this trick doesn't stop at puzzles and Olympiad problems - it also shows up in real research. Arguably, the recent breakthrough for the sofa problem used this idea, and I've...

Hi, it's Choo Ray. Recently, I visited a SIMO National Team training session to soak in the atmosphere and meet some young friends. The session was about sequences and had quite a few interesting questions, so it was a pity that attendance was low (coincidentally many students were involved in the National Olympiad in Informatics). Today I would like to highlight a particular question that intrigued me and discuss some variations. Full Score For those of you looking for a challenge, I will list all variations here. Sequences, Example 2.3 Let $a_1,a_2,...$ and $b_1,b_2,...$ and $c_1,c_2,...$ be three arbitrary infinite sequences of positive integers. Prove that there exist different indices, $r,s,t$ such that $a_r \ge a_s \ge a_t$ and $b_r \ge b_s \ge b_t$ and $c_r \ge c_s \ge c_t$. Variation 1: Distinct positive integers Let $a_1,a_2,...$ and $b_1,b_2,...$ and $c_1,c_2,...$ be three arbitrary infinite sequences of distinct positive integers. Prove that there exist diffe...