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Fixed points of composition of inversions

Source: KoMaL A. 714

May 21, 2023
geometryanalysiscombinatorial geometry

Problem Statement

Consider n≥2n \ge 2 pairwise disjoint disks D1,D2,…,DnD_1,D_2,\ldots,D_n on the Euclidean plane. For each k=1,2,…,nk=1,2,\ldots,n, denote by fkf_k the inversion with respect to the boundary circle of DkD_k. (Here, fkf_k is defined at every point of the plane, except for the center of DkD_k.) How many fixed points can the transformation fn∘fn−1∘…∘f1f_n\circ f_{n-1}\circ\ldots\circ f_1 have, if it is defined on the largest possible subset of the plane?
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