PUMaC 2024 · 代数（A 组） · 第 3 题

PUMaC 2024 — Algebra (Division A) — Problem 3

T consists of a single branch from 0 to 1 in the complex plane. This branch then splits into 1 ◦ two branches at the endpoint which each form a 135 angle with the branch in T , and each 1 1 √ branch has length . This process is repeated, so that from each terminal branch in T , we n 2 1 ◦ √ form two more branches at angles 135 with the length. Let L be the collection of the n 2 n − 1 2 endpoints of the tree T . If multiple terminal branches end at the same point, then that n point is counted multiple times in L . Shown below is T for k = 1 , 2 , 3 with dots at the points n k 2 in L . Find the sum of ℓ over all points ℓ in L . k 10 (a) T and L (b) T and L (c) T and L 1 1 2 2 3 3 1

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