HMMT November 2024 #0l41

HMMT November 2024Hard

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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.

TangentsAngle chasing

Problem

Problem:

Derek is bored in math class and is drawing a flower. He first draws 8 points $A_{1}, A_{2}, \ldots, A_{8}$ equally spaced around an enormous circle. He then draws 8 arcs outside the circle where the $i$ th arc for $i=1,2, \ldots, 8$ has endpoints $A_{i}, A_{i+1}$ with $A_{9}=A_{1}$ , such that all of the arcs have radius 1 and any two consecutive arcs are tangent. Compute the perimeter of Derek's 8-petaled flower.

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