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2023 SMT Guts Round 8 p22-24 - Stanford Math Tournament

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August 31, 2023
Stanford Math Tournamentgeometrynumber theorycombinatoricsalgebra

Problem Statement

p22. Consider the series {An}n=0∞\{A_n\}^{\infty}_{n=0}, where A0=1A_0 = 1 and for every n>0n > 0, An=A[n2023]+A[n20232]+A[n20233],A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]}, where [x][x] denotes the largest integer value smaller than or equal to xx. Find the (202332+20)(2023^{3^2}+20)-th element of the series.
p23. The side lengths of triangle △ABC\vartriangle ABC are 55, 77 and 88. Construct equilateral triangles △A1BC\vartriangle A_1BC, △B1CA\vartriangle B_1CA, and △C1AB\vartriangle C_1AB such that A1A_1,B1B_1,C1C_1 lie outside of △ABC\vartriangle ABC. Let A2A_2,B2B_2, and C2C_2 be the centers of △A1BC\vartriangle A_1BC, △B1CA\vartriangle B_1CA, and △C1AB\vartriangle C_1AB, respectively. What is the area of △A2B2C2\vartriangle A_2B_2C_2?
p24. There are 2020 people participating in a random tag game around an 2020-gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the 2020-gon (no matter where they were at the beginning). If there are currently 1010 taggers, let EE be the expected number of untagged people at the end of the next round. If EE can be written as ab\frac{a}{b} for a,ba, b relatively prime positive integers, compute a+ba + b.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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