Posts

Look, I didn't bash a geometry problem!

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This is Glen again. There's been a lack of regular olympiad content recently, and since no one posted this week I thought I'd attempt a problem and write about my thought process. There was some discussion recently on the retirees' Discord server about APMO, so I figured I'd look at the most recent iteration, and since I wanted to be able to solve the problem in a reasonable amount of time, I attempted Q1. Here's the problem: ( APMO 2024/1 ) Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear. Preliminary thoughts First and foremost, here's a diagram: Sorry about th...
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Introduction to UFDs

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(This is Glen.) I was sorting through the LaTeX files on my computer and unearthed an old set of solutions to a mysterious problem set, dated 2020. After a bit of digging in some old Discord servers, I found out that these were solutions to one of Zhao Yu's sets for some RI training, which I had presumably crashed because it was online (thanks to Covid) and I was too free or something. Anyway, this file contained a lengthy introduction to UFDs, which I had recently learnt about in uni and had used to overkill a couple of problems in the set. This is, I think, quite suitable for a blog post, so here we are. The fundamental theorem of arithmetic As a warmup, let's think about something we learn about in primary school (well, at least I remember learning about this in primary school but I am old so this may no longer be the case): the unique prime factorisation of integers. (Fundamental theorem of arithmetic) Each integer $n>1$ can be written uniquely as $n=p_1\cdots p_k$, wher...
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A one-line lower bound for prime counts

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 (Zhao Yu here). Today someone spoke about this short proof of a lower bound of the number of primes, which I found it too nice to not share.
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A practical guide to the group law

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(David here.) We've had at least two articles on elliptic curves so far: Zhao Yu's article talks about some neat applications across various areas of math, while Dylan's article focuses on motivating the group law. Recently, I found out that the group law can be used to tackle actual Olympiad geometry problems (!), so I thought I might write a guide as to how.
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Algebra ∩ Geometry via Desargues' Involution Theorem

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(David here.) Recently a problem showed up on the problems feed in the SIMO Retirees' Discord server that made me take a second look at Desargues' Involution Theorem, and I realized it was actually the perfect starting point to discuss the interplay between geometry and algebra!
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Dynamics with USATST '25 P2

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(USATST '25 P2) Let $a_1,a_2,\dots,$ and $b_1,b_2,\dots,$ be sequences of real numbers for which $a_1 > b_1$ and $a_{n+1} = a_n^2-2b_n, b_{n+1}=b_n^2-2a_n$ for all positive integers $n$. Prove that $a_1,a_2,\dots,$ is eventually increasing (i.e. there exists a positive integer $N$ for which $a_k < a_{k+1}$ for all $k>N$). (Dylan here.) I will be writing this post as I am trying to solve the above problem.
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Unsolved Problems VS Olympiad NT

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(Drew here.) I've spent a lot of time browsing random math pages on Wikipedia  (blue link). Sometimes I find things which are actually useful in olympiads, but a lot of the time there's this void of open problems. Of these open problems, some were only solved recently, and they can be used to blast olympiad problems.
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