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Turkey NMO 2008 1st Round - P13 (Geometry)

Source:

August 25, 2012
geometrytrigonometry

Problem Statement

Let ABCABC be a triangle such that angle CC is obtuse. Let D[AB]D\in [AB] and [DC][BC][DC]\perp [BC]. If m(ABC^)=αm(\widehat{ABC})=\alpha, m(BCA^)=3αm(\widehat{BCA})=3\alpha, and ACAD=10|AC|-|AD|=10, what is BD|BD|?
<spanclass=latexbold>(A)</span> 10<spanclass=latexbold>(B)</span> 14<spanclass=latexbold>(C)</span> 18<spanclass=latexbold>(D)</span> 20<spanclass=latexbold>(E)</span> 22 <span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 18 \qquad<span class='latex-bold'>(D)</span>\ 20 \qquad<span class='latex-bold'>(E)</span>\ 22