2021 WSMO Team Round Problems/Problem 4

Problem

Consider a triangle $A_1B_1C_1$ satisfying $A_1B_1=3,B_1C_1=3\sqrt{3},A_1C_1=6$ . For all successive triangles $A_nB_nC_n$ , we have $A_nB_nC_n\sim B_{n-1}A_{n-1}C_{n-1}$ and $A_n=B_{n-1},C_n=C_{n-1}$ , where $A_nB_nC_n$ is outside of $A_{n-1}B_{n-1}C_{n-1}$ . Find the value of $\left(\sum_{i=1}^{\infty}[A_iB_iC_i]\right)^2,$ where $[A_iB_iC_i]$ is the area of $A_iB_iC_i$ .

Proposed by pinkpig

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2021 WSMO Team Round Problems/Problem 4

Problem

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