MathDB
Bounds of the last point of the operation

Source: CIIM 2022 - Problem 3

September 19, 2023
geometrycomplex number geometrymaximum and minimumvector geometry

Problem Statement

Danielle draws a point OO on the plane and a set of points P={P0,P1,…,P2022}\mathcal P = \{P_0, P_1, \ldots , P_{2022}\} such that ∠P0OP1=∠P1OP2=⋯=∠P2021OP2022=α,0<α<π,\angle{P_0OP_1} = \angle{P_1OP_2} = \cdots = \angle{P_{2021}OP_{2022}} = \alpha, \hspace{5pt} 0 < \alpha < \pi,where the angles are measured counterclockwise and for 0≤n≤20220 \leq n \leq 2022, OPn=rnOP_n = r^n, where r>1r > 1 is a given real number. Then, obtain new sets of points in the plane by iterating the following process: given a set of points {A0,A1,…,An}\{A_0, A_1, \ldots , A_n\} in the plane, it is built a new set of points {B0,B1,…,Bn−1}\{B_0, B_1, \ldots , B_{n-1}\} such that AkAk+1BkA_kA_{k+1}B_k is an equilateral triangle oriented clockwise for 0≤k≤n−10 \leq k \leq n - 1. After carrying out the process 20222022 times from the set PP, Danielle obtains a single point XX. If the distance from XX to point OO is dd, show that (r−1)2022≤d≤(r+1)2022.(r-1)^{2022} \leq d \leq (r+1)^{2022}.
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