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2012 DMM Tiebreaker Round - Duke Math Meet

Source:

October 2, 2023
DMMalgebrageometrytrigonometry

Problem Statement

p1. An 88-inch by 1111-inch sheet of paper is laid flat so that the top and bottom edges are 88 inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold?
p2. Triangle ABCABC is equilateral, with AB=6AB = 6. There are points DD, EE on segment AB (in the order AA, DD, EE, BB), points FF, GG on segment BCBC (in the order BB, FF, GG, CC), and points HH, II on segment CACA (in the order CC, HH, II, AA) such that DE=FG=HI=2DE = F G = HI = 2. Considering all such configurations of DD, EE, FF, GG, HH, II, let A1A_1 be the maximum possible area of (possibly degenerate) hexagon DEFGHIDEF GHI and let A2A_2 be the minimum possible area. Find A1−A2A_1 - A_2.
p3. Find tan⁡π7tan⁡2π7tan⁡3π7\tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7}
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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