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Local to global: ISL 2018 C7

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(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...
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Series acceleration and zeta(2)

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(This is Yan Sheng.) Today's post started with this question: IMC 2015 Q6 : Prove that $\displaystyle\sum_{n=1}^\infty\frac1{\sqrt n(n+1)}<2$. The interested reader can pause here to try it out, while I helpfully fill the next few sentences with flavour text to delay spoilers. One of my favourite hobbies is to cosplay as Euler. No, not literally (though I'm sure some of you freaks who read this blog would love to see it), but by trying to manipulate infinite series in all sorts of fun ways. I've already written here before about hand computation and zeta(2) , and this post will be more of the same mucking around. Theodorus's constant For today's problem, seeing that the terms are of order $n^{-3/2}$, the first instinct should be to compare with a telescoping series whose tail sums are of order $n^{-1/2}$. Indeed, it's completely routine to check that$$2\left(\frac1{\sqrt n}-\frac1{\sqrt{n+1}}\right)-\frac1{\sqrt n(n+1)}=\frac{(\sqrt{n+1}-\sqrt n)^2}...
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Who is Alice, and why do her coins give multiples of everything?

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INT. CLASSROOM SOMEWHERE IN NUS — LATE AFTERNOON Close-up of an open copy of Bott-Tu on the desk, atop pages of largely illegible scratchwork. Slow zoom out and pan up to a duo in front of the whiteboard. In the far corner of the frame, a third becomes visible, seated, lightly patting one's own tummy.  "Let's write an IMO problem," said one.  "Sure," said the other. "Get back to real math," said the third. But it was too late: the juices of creativity and mathematical delusion had begun flowing, unyielding to all forces of nature (except perhaps dinnertime).  "Let's start with a grid of numbers." A grid was drawn on the board, but no numbers were written. Indeed, it was because they did not yet know what numbers belonged in the grid. "Integers? Rationals? Real numbers?" "Maybe some number theory thing... let's stick with integers first." "Okay." The grid remained empty, like an empty stomach. "Ho...
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A grid arrangement problem

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(David here.) I'm back with another interesting problem - this time, it's about arranging numbers in a grid.
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Taylor series in number theory

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Glen here. At some point about a year ago, I scrolled through the blogger interface and found an old draft by Dylan, a skeletal outline of a post featuring a problem and a few section headings. The problem happened to be one I recognised: Show that $2^{2000}$ divides the numerator of $2+\frac{2^2}2 + \frac{2^3}3 + \cdots + \frac{2^{2024}}{2024}$ when expressed as a simplified improper fraction. I remembered first seeing this as a bonus problem in some old SIMO National Team set, when I'd looked at it, had absolutely no idea how to approach it, and promptly given up. But this time, having learnt some more math in the intervening years, I did  have an idea of what to do, and ended up solving it it in my head during some seminar that I wasn't quite following. The solution turned out to be pretty cool, so I made a note to myself to write a blog post about it at some point if Dylan's post never manifested. I then promptly forgot about it. Fast forward to this week. Some of the i...
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A cool problem about function composition

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(David here.) It's been a while since the last blog post, so I thought I'd share a problem that I found interesting recently.
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SMO Open 2025 ??% Speedrun

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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...
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