SIMO Retiree Blog

(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. Today I'm going to present a fun curiosity that I saw on Mathologer (luckily somebody did the 'reading old papers' bit for me and spared me having to decipher old French) just because I think it ought to be better known. We'll then see where it takes us. In 1867 an Austrian military engineer by the name of M. E. Lill published the following method of finding real roots of polynomial equations:  Let $p(x)= a_nx^n + a_{n-1}x^{n-1} + \cdots a_0$ be a polynomial with real coefficients. Starting at the origin $O$ in the coordinate plane, face the direction of the positive $x-$axis. Walk the signed distance $a_n$ (this means that if $a_n$ is negative then walk backwards) and turn $90$ degrees cou...

(This is Glen.) I was sorting through my stuff and found a piece of paper from when I tried last year's NTST problems, which I presumably saved because I thought I might eventually want to write a post about it. So here I am, writing a post so I can safely dump this piece of paper into my recycling bin. One functional inequality Here's the relevant problem: ( ISL 2023 A4 )  Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \left(f(x) + f(y)\right) \geqslant \left(f(f(x)) + y\right) f(y)\] for every $x, y \in \mathbb R_{>0}$. This was NTST 2024 Day 1 Q2, by the way. To avoid spoilers, I'm going to put a very crudely censored version of my working, and then reveal each bit one by one, in the order that I think I wrote everything in. Here we go: Anyway, let's begin. Rectangle #1 Yay, it's the problem statement! Rectangle #2 Here, I tried some simple substitutions. The...