SIMO Retiree Blog

 (This is Glen.) At some point, Ker Yang wrote a post on Reim's theorem, which he used to solve two past year IMO problems. I remember commenting to him that I had solved neither of them with Reim (and no, I did not bash them). Later on, I tried reconstructing my solution to IMO 2017/4, and I noticed something interesting that made me find another (slightly weird) solution that (I think) isn't on AoPS. So that's what I'll be writing about today. First, the problem: ( IMO 2017/4 ) Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$...

(This is Yan Sheng.) In my mind, J. Kenji López-Alt is the greatest nerdsniper chef: he has made claims about cooking methods and techniques, based on experimental evidence, which have become the inspiration for later theoretical research. For instance, on the subject of grilling meat , he writes that "...flipping steak repeatedly during cooking can result in a cooking time about 30% faster than flipping only once"; this was confirmed by a model of Thiffeault (2022) (here are some slides from a talk ). Today's post, however, is about onions: Question : What is the optimal way to cut an onion such that the size variation among the diced pieces is minimised? After removing the ends and halving the onion, we can view it as approximately a half-cylinder with concentric layers. This reduces the dimension of the problem, so now we're cutting concentric semicircles: (In the following diagrams, I'm viewing the arcs themselves as the layers, and not the spaces bet...