PUMaC 2024 · 几何（A 组） · 第 7 题

PUMaC 2024 — Geometry (Division A) — Problem 7

The following is the construction of the twindragon fractal. I is the solid square region with 0 1 1 1 1 vertices at (0 , 0) , ( , ) , (1 , 0) , ( , − ). Recursively, the region I consists of two copies of n +1 2 2 2 2 ◦ I : one copy which is rotated 45 counterclockwise around the origin and scaled by a factor of n 1 ◦ √ , and another copy which is also rotated 45 counterclockwise around the origin and scaled 2 1 1 1 √ by a factor of and then translated by ( , − ). We have displayed I and I below. Let I 0 1 ∞ 2 2 2 be the limiting region of I , I , . . . . The area of the smallest convex polygon which encloses 0 1 a I can be written as for relatively prime positive integers a and b . Find a + b . ∞ b 1 y y 1 1 x x 1 I I 0 1 − 1 − 1

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