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Part of 2022 Taiwan TST Round 1

Problems(1)

Mingmingsan playing with an equilateral triangle

Source: 2022 Taiwan TST Round 1 Independent Study 2-C

3/17/2022
Let △P1P2P3\triangle P_1P_2P_3 be an equilateral triangle. For each n≥4n\ge 4, Mingmingsan can set PnP_n as the circumcenter or orthocenter of △Pn−3Pn−2Pn−1\triangle P_{n-3}P_{n-2}P_{n-1}. Find all positive integer nn such that Mingmingsan has a strategy to make PnP_n equals to the circumcenter of △P1P2P3\triangle P_1P_2P_3.
Proposed by Li4 and Untro368.
combinatoricsgeometrycircumcircleming