HMMT February 2020 #0ke5

HMMT February 2020Hard

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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.

Problem

Problem:

Alan draws a convex 2020-gon $\mathcal{A}=A_{1} A_{2} \cdots A_{2020}$ with vertices in clockwise order and chooses 2020 angles $\theta_{1}, \theta_{2}, \ldots, \theta_{2020} \in (0, \pi)$ in radians with sum $1010 \pi$ . He then constructs isosceles triangles $\triangle A_{i} B_{i} A_{i+1}$ on the exterior of $\mathcal{A}$ with $B_{i} A_{i}=B_{i} A_{i+1}$ and $\angle A_{i} B_{i} A_{i+1}=\theta_{i}$ . (Here, $A_{2021}=A_{1}$ .) Finally, he erases $\mathcal{A}$ and the point $B_{1}$ . He then tells Jason the angles $\theta_{1}, \theta_{2}, \ldots, \theta_{2020}$ he chose. Show that Jason can determine where $B_{1}$ was from the remaining 2019 points, i.e. show that $B_{1}$ is uniquely determined by the information Jason has.