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2023 SMT Guts Round 5 p13-15 - Stanford Math Tournament

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August 31, 2023
Stanford Math Tournamentgeometrycombinatoricsnumber theory

Problem Statement

p13. Let △ABC\vartriangle ABC be an equilateral triangle with side length 11. Let the unit circles centered at AA, BB, and CC be ΩA\Omega_A, ΩB\Omega_B, and ΩC\Omega_C, respectively. Then, let ΩA\Omega_A and ΩC\Omega_C intersect again at point DD, and ΩB\Omega_B and ΩC\Omega_C intersect again at point EE. Line BDBD intersects ΩB\Omega_B at point FF where FF lies between BB and DD, and line AEAE intersects ΩA\Omega_A at GG where GG lies between AA and EE. BDBD and AEAE intersect at HH. Finally, let CHCH and FGFG intersect at II. Compute IHIH.
p14. Suppose Bob randomly fills in a 45×4545 \times 45 grid with the numbers from 11 to 20252025, using each number exactly once. For each of the 4545 rows, he writes down the largest number in the row. Of these 4545 numbers, he writes down the second largest number. The probability that this final number is equal to 20232023 can be expressed as pq\frac{p}{q} where pp and qq are relatively prime positive integers. Compute the value of pp.
p15. ff is a bijective function from the set {0,1,2,...,11}\{0, 1, 2, ..., 11\} to {0,1,2,...,11}\{0, 1, 2, ... , 11\}, with the property that whenever aa divides bb, f(a)f(a) divides f(b)f(b). How many such ff are there? A bijective function maps each element in its domain to a distinct element in its range.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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