Difference between revisions of "2021 WSMO Team Round Problems/Problem 4"
(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== | ==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 <math>A_nB_nC_n\sim B_{n-1}A_{n-1}C_{n-1}</math> and <math>A_n=B_{n-1},C_n=C_{n-1}</math>, where <math>A_nB_nC_n</math> is outside of <math>A_{n-1}B_{n-1}C_{n-1}</math>. Find the value of<cmath>\left(\sum_{i=1}^{\infty}[A_iB_iC_i]\right)^2,</cmath>where <math>[A_iB_iC_i]</math> is the area of <math>A_iB_iC_i</math>. | + | 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 <math>A_nB_nC_n\sim B_{n-1}A_{n-1}C_{n-1}</math> and <math>A_n=B_{n-1},C_n=C_{n-1}</math>, where <math>A_nB_nC_n</math> is outside of <math>A_{n-1}B_{n-1}C_{n-1}</math>. Find the value of <cmath>\left(\sum_{i=1}^{\infty}[A_iB_iC_i]\right)^2,</cmath> where <math>[A_iB_iC_i]</math> is the area of <math>A_iB_iC_i</math>. |
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| + | ''Proposed by pinkpig'' | ||
==Solution== | ==Solution== | ||
Latest revision as of 21:16, 15 December 2023
Problem
Consider a triangle 
satisfying 
. For all successive triangles 
, we have 
and 
, where is outside of 
. Find the value of ![\[\left(\sum_{i=1}^{\infty}[A_iB_iC_i]\right)^2,\]](http://latex.artofproblemsolving.com/8/b/8/8b874d899b2587637d3789ce1391ca043f9cdfbd.png)
where ![$[A_iB_iC_i]$](http://latex.artofproblemsolving.com/3/b/d/3bd50946fbcfddd795feec3f5c91ceb0b652fa04.png)
is the area of 
.
Proposed by pinkpig
