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2024 AMC 12 B #0l03

2024 AMC 12 BHard

From module Angle Chasing Mastery

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Understand the scale
Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.
RotationTriangle trigonometryTriangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleAngle chasingDistance chasing

Problem

Equilateral △ABC\triangle ABC with side length 1414 is rotated about its center by angle θ\theta, where 0<θ≤60∘0 < \theta \le 60^\circ, to form △DEF\triangle DEF. See the figure. The area of hexagon ADBECFADBECF is 91391\sqrt{3}. What is tan⁡θ\tan \theta? figure (A) 35\frac{3}{5} (B) 5311\frac{5\sqrt{3}}{11} (C) 45\frac{4}{5} (D) 1113\frac{11}{13} (E) 7313\frac{7\sqrt{3}}{13}

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