SIMO Retiree Blog

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...

(USATST '25 P2) Let $a_1,a_2,\dots,$ and $b_1,b_2,\dots,$ be sequences of real numbers for which $a_1 > b_1$ and $a_{n+1} = a_n^2-2b_n, b_{n+1}=b_n^2-2a_n$ for all positive integers $n$. Prove that $a_1,a_2,\dots,$ is eventually increasing (i.e. there exists a positive integer $N$ for which $a_k < a_{k+1}$ for all $k>N$). (Dylan here.) I will be writing this post as I am trying to solve the above problem.

Hi, Choo Ray here. Before IMO 2024, the Singapore team had a joint friendly contest with Iran and Taiwan (spoilers ahead). As I found out, it takes quite a lot of work to run a contest. We wanted to simulate the IMO format in terms of topic diversity and difficulty but our shortlist lacked depth in certain topics.

 (Glen here.) I had the privilege of being a Coordinator at this year's IMO, which, along with some extra perks such as a lot of free food and being able to witness Singapore's 3rd-best ever team placement (!), came with the benefit of seeing this year's IMO problems a couple of days before everyone else. This also meant that I had a couple of days to make predictions about how each problem would be received, and I turned out to be wildly wrong about one problem in particular: Problem 3. I'd found this a lot easier than its single-digit solve-rate would suggest (this took me less than an hour without paper), and so I figured that I should write something about my thought process behind this problem. For obvious reasons, I don't have an actual record of solving the problem, but everything has been reconstructed to the best of my memory. The problem I first saw this problem on the Saturday before the IMO. There was a nice fancy dinner for Coordinators and Leaders, and...

(Jit here). Today I want to explain how to prove Skolem-Mahler-Lech using the p-adic numbers, and explain some sort of generalization in the area of algebraic/arithmetic dynamics, known as the Dynamical Mordell-Lang conjecture, which is still an open problem.