Bay Area Mathematical Olympiad #0k4o

Bay Area Mathematical OlympiadHard

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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:

To dissect a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:

An integer-ratio right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides $3, 4, 5$ is an integer-ratio right triangle, and so is a triangle with sides $\frac{5}{2} \sqrt{3}, 6 \sqrt{3}, \frac{13}{2} \sqrt{3}$ . On the other hand, the right triangle with sides $\sqrt{2}, \sqrt{5}, \sqrt{7}$ is not an integer-ratio right triangle.

Determine, with proof, all integers $n$ for which it is possible to completely dissect a regular $n$ -sided polygon into integer-ratio right triangles.