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...

(David here.) In this post, I go through a surprisingly tricky functional equation that appeared on the 2018 edition of the IMO Revenge, a contest where the contestants made problems for their trainers. The problem (IMO Revenge 2018/4) Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that $$f(x)^2-f(y)^2=f(x+y)\cdot f(x-y)$$ for all $x,y\in \mathbb{Q}$. Fun fact - I was actually at that IMO as an observer! I had good memories of attempting the test, solving problem 3 and meeting the contestant-proposer (who later went on to propose an actual IMO Q3). Initial observations Clearly, $f(x) = x$ works. Furthermore, the equation is homogeneous in $f$ - if $f$ works then so must $cf$. It's also cheap to get that $f(0) = 0$ and $f(-x) = -f(x)$. This was roughly where I ran out of cheap things to find - I didn't manage to get any more special values or any standard properties (like injectivity or surjectivity). Some progress When you get stuck on a functional equati...

(This is Yan Sheng.) Last time we studied a family of inequalities involving absolute values on linear polynomials, and asked the question of how lazy we can be to prove inequalities of that form. This week, we will be applying the same approach to a different family: 3-variable homogeneous symmetric inequalities of low degree. A polynomial $P(x,y,z)$ in 3 variables is called homogeneous of degree $d$ if each of its terms is of degree $d$, and symmetric if $P(x,y,z)=P(x',y',z')$ for all permutations $x',y',z'$ of $x,y,z$. Inequalities involving homogeneous symmetric polynomials include the AM-GM inequality$$\left(\frac{x+y+z}3\right)^3\ge xyz,$$and the Schur inequality$$x^r(x-y)(x-z)+y^r(y-x)(y-z)+z^r(z-x)(z-y)\ge0$$for integers $r\ge0$. For the rest of this post, write $\mathcal P^+_d$ for the family of all homogeneous symmetric polynomials $P(x,y,z)$ of degree $d$ such that $P(x,y,z)\ge0$ holds for all $x,y,z\ge0$. Our Main Problem is to describe $\math...