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Turkey NMO 2007 1st Round - P23 (Algebra)

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October 5, 2012
geometry

Problem Statement

A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are middle one-third segments of the sides of the polygon. After repeating these steps for infinite times, what is the area of the new shape?
<spanclass=′latex−bold′>(A)</span> 123<spanclass=′latex−bold′>(B)</span> 38<spanclass=′latex−bold′>(C)</span> 310<spanclass=′latex−bold′>(D)</span> 143<spanclass=′latex−bold′>(E)</span> None of the above <span class='latex-bold'>(A)</span>\ \dfrac {1}{2\sqrt 3} \qquad<span class='latex-bold'>(B)</span>\ \dfrac {\sqrt 3}{8} \qquad<span class='latex-bold'>(C)</span>\ \dfrac {\sqrt 3}{10} \qquad<span class='latex-bold'>(D)</span>\ \dfrac {1}{4\sqrt 3} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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