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Maximum number of points place-able (KJMO 1999 P7)

Source: 1999 KJMO P6

June 30, 2024
pointsgeometrycombinatoricsKJMO

Problem Statement

A0B,A0CA_0B, A_0C rays that satisfy ∠BA0C=14∘\angle BA_0C=14^{\circ}. You are to place points A1,A2,...A_1, A_2, ... by the following rules.
Rules (1) On the first move, place A1A_1 on any point on A0BA_0B(except A0A_0).
(2) On the n>1n>1th move, place AnA_n on A0BA_0B iff An−1A_{n-1} is on A0CA_0C, and place AnA_n on A0CA_0C iff An−1A_{n-1} is one A0BA_0B. AnA_n must be place on the point that satisfies An−2Ann−1=An−1AnA_{n-2}A_n{n-1}=A_{n-1}A_n.
All the points must be placed in different locations. What is the maximum number of points that can be placed?
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