SIMO Retiree Blog

(This is Glen.) There hasn't been a post written for this week, so I figured I'd scroll through AoPS and try a problem that looked interesting. This ended up being: ( ISL 2018 C7 ) Consider $2018$ pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular edges that meet at vertices . Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doing so for each circle, every vertex is coloured twice - once for each of the two circle that cross at that point. If the two colours agree at a vertex, then it is assigned that colour; otherwise, it becomes yellow. Show that, if some circle contains at least $2061$ yellow points, then the vertices of some region are all yellow. In theory, the 2018 shortlist was the one that I had early access to (since I was an Observer in 2019), but I don't remember trying this problem. I was p...

(Guest author Andrew here!) A good exercise for Olympiad students is to prove the following: A real polynomial $f(x)\in \R[x]$ which is everywhere non-negative ( psd , short for positive semi-definite) must be a sum of squares ( sos ).  This is in some sense an algebraic witness to the analytic property of being non-negative, and is an example of what is known as a Positivstellensatz . One might reasonably ask whether this extends to the case of multiple variables, and it turns out the answer is no .  Consider the two variable polynomial $g(x,y)=x^4y^2+x^2y^4-3x^2y^2+1$ (the Motzkin polynomial ). Non-negativity follows from the AM-GM inequality. But how can we show that it isn't a sum of squares? We now introduce an object known as the Newton polytope. Given a polynomial $f$,  To every monomial with non-zero coefficient, assign a point whose $i^{th}$ coordinate is the degree of the $i^{th}$ variable in that monomial, e.g. assign to $x_1^n$ the point $(n, 0)$. The Newton p...

(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...

(Andrew here.)  In research, it often happens that people rediscover results lost to the sands of time, which always makes me wonder what gemstones have been lost and are waiting to be rediscovered. It also means that a lot of surprising things can be learned from reading old papers.

(Jeck here.) Consider an arrangement of $n$ lines on the plane, which do not necessarily need to be in general position. There are many combinatorial problems one can explore with such configurations: If no three lines are concurrent, the number of regions into which these lines divide the plane is given by $1+\binom{n+1}{2}$. If not all lines are concurrent, there is always at least one intersection point that is shared by exactly two lines.

(This is Glen.) Some time ago, Etienne posted about  this problem on rectangle tilings , to which this was my reaction in our blog Discord channel: Glen declares this to be his favourite combi problem. I, too, think that this problem deserves an article of its own, and so this is my attempt.

(Etienne here.) The date was 24 October 2021. It was late at night, and I was just about to tuck myself in. After all, the next day was the A-Level Physics practical! I needed to be well-rested (spoiler alert - I actually didn't). I was lying comfortably and about to bid a temporary farewell to the waking world, when suddenly, out of nowhere, I was attacked by a sudden thought: For what $r$ could I tile a square with rectangles whose sides had ratio $r$?

(Dylan here.) Today, I'll be solving a (pseudorandomly chosen) combinatorics problem. But instead of just explaining the solution, I will detail my messy thought process as I work through it in real time.  After solving the problem, I will do a brief meta-commentary, and try to abstract the problem. Finally, for the more mathematically mature, I will talk about what possibly lies beyond it: various extensions and related areas. Let's begin. 1     The Problem (APMO 2023 P1) Let $n\geq 5$ be an integer. Consider $n$ squares with side lengths $1,2,\dots,n$ respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly $2$ other squares.  1.1     First Thoughts In no particular order: This is a  construction  problem.  The orientation of the ...