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Showing posts from September, 2023

SMO Open 2023 ??% Speedrun

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(This is Glen.) I'm doing this since I'm relatively free at the moment (it's still summer vacation) and I've been looking for an excuse to try these problems anyway. I'll give myself 4 hours (which I think is the duration of the competition?), type everything I'm thinking of in bullet points and add headers to make things more readable at the end. I'm not writing out solutions properly because the focus is on the thought process rather than the actual solution but the intent is that the solution is still discernable if you follow everything I write down. All in all, the time saved in not having to write things nicely should cancel out with the extra time I spend typing out all of my thoughts. Anyway, here I go. Let's hope I don't embarrass myself. Problem 1 In a scalene triangle ABCABC with centroid GG and circumcircle ω\omega centred at OO, the extension of AGAG meets ω\omega at MM; lines ABAB and CMCM intersect at PP; and lines ACAC and BMBM i...

Combi Solving: APMO 2023 P1

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(Dylan here.) Today, I'll be solving a (pseudorandomly chosen) combinatorics problem. But instead of just explaining the solution, I will detail my messy thought process as I work through it in real time.  After solving the problem, I will do a brief meta-commentary, and try to abstract the problem. Finally, for the more mathematically mature, I will talk about what possibly lies beyond it: various extensions and related areas. Let's begin. 1     The Problem (APMO 2023 P1) Let n5n\geq 5 be an integer. Consider nn squares with side lengths 1,2,,n1,2,\dots,n respectively. The squares are arranged in the plane with their sides parallel to the xx and yy axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly 22 other squares.  1.1     First Thoughts In no particular order: This is a  construction  problem.  The orientation of the ...

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