SIMO Retiree Blog

Glen & Sheldon here. At last, our schedules aligned enough for us to try this year's SMO Open together. Here's an outline of our thought process while solving the problems, featuring some screenshots from our Cocreate whiteboard. Problem 1 In the triangle $ABC$, $\angle B>90^\circ$, the incircle touches the sides $BC$ and $CA$ at $D$ and $E$, respectively. The lines $ED$ and $AB$ intersect at $P$. The incircle of the triangle $AEP$ touches the sides $PE$ and $AP$ at $D_1$ and $E_1$, respectively. The lines $E_1D_1$ and $AE$ intersect at $P_1$. Suppose $P,C,E,B$ are concyclic. Prove that $BE$ is parallel to $PP_1$. Weird-looking problem. The concyclic condition means $\angle ABC = \angle AED$. Maybe it could be less weird if we write it in terms of the angles of the triangle, $\angle ABC = 90^\circ + \frac{\angle ACB}2$. Actually, the equal angles imply $\triangle ABC \sim \triangle AEP$. The parallel condition is equivalent to $\frac{AB}{AE} = \frac{AP}{AP_1}$, and the s...

(Glen here.) Recently, there was some discussion about this year's IOI in a group chat, and Ker Yang remarked that the first task, " Souvenirs ", was basically a math question. Of course, I had to have a look. The original description is pretty long, so here's a rewriting in a more math olympiad style: Sheldon is buying french fries from a fast food restaurant, which sells fries in $N$ different sizes (with packagings labelled from largest to smallest from $0$ to $N-1$), with integer costs $P_0 > P_1 > \ldots > P_{N-1} > 0$. To buy fries, Sheldon can pay $\$M$ where $M$ is a positive integer, and then define a sequence $i_1,i_2,\ldots$ as follows: $i_1$ is the smallest integer such that $P_{i_1}\le M$, and in general, $i_k$ is the smallest integer $>i_{k-1}$ such that $P_{i_k} \le M - P_{i_1} - \cdots P_{i_{k-1}}$. If no such $i_k$ can be defined, the sequence terminates. Then, Sheldon receives the fries with labels $i_1,i_2,\ldots$ along with his chang...