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Great combination of geometry and analytic arguments

Source: Bulgaria MO Regional round 2024, 12.3

February 13, 2024
geometry

Problem Statement

Let A0B0C0A_0B_0C_0 be a triangle. For a positive integer n≥1n \geq 1, we define AnA_n on the segment Bn−1Cn−1B_{n-1}C_{n-1} such that Bn−1An:Cn−1An=2:1B_{n-1}A_n:C_{n-1}A_n=2:1 and Bn,CnB_n, C_n are defined cyclically in a similar manner. Show that there exists an unique point PP that lies in the interior of all triangles AnBnCnA_nB_nC_n.
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