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SMT 2011 Geometry # 6

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September 2, 2022
geometry

Problem Statement

Two parallel lines ℓ1\ell_1 and ℓ2\ell_2 lie on a plane, distance dd apart. On ℓ1\ell_1 there are an infinite number of points A1,A2,A3,...A_1, A_2, A_3, ... , in that order, with AnAn+1=2A_nA_{n+1} = 2 for all nn. On ℓ2\ell_2 there are an infinite number of points B1,B2,B3,...B_1, B_2, B_3,... , in that order and in the same direction, satisfying BnBn+1=1B_nB_{n+1} = 1 for all nn. Given that A1B1A_1B_1 is perpendicular to both ℓ1\ell_1 and ℓ2\ell_2, express the sum ∑i=1∞∠AiBiAi+1\sum_{i=1}^{\infty} \angle A_iB_iA_{i+1} in terms of dd. https://cdn.artofproblemsolving.com/attachments/c/9/24b8000e19cffb401234be010af78a6eb67524.png
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