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2021 WSMO Team Round Problems/Problem 4

Revision as of 22:08, 15 December 2023 by Pinkpig (talk | contribs) (Created page with "==Problem== Consider a triangle <math>A_1B_1C_1</math> satisfying <math>A_1B_1=3,B_1C_1=3\sqrt{3},A_1C_1=6</math>. For all successive triangles <math>A_nB_nC_n</math>, we have...")
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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 $A_iB_iC_i$.

Solution

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