SIMO Retiree Blog

(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...