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Turkey NMO 2007 1st Round - P33 (Geometry)

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October 5, 2012
geometry

Problem Statement

The tangent lines from the point AA to the circle CC touches the circle at MM and NN. Let PP a point on [AN][AN]. Let MPMP meet CC at QQ. Let MNMN meet the line through PP and parallel to MAMA at RR. If ∣MA∣=2|MA|=2, ∣MN∣=3|MN|=\sqrt 3, and QR∥ANQR \parallel AN, what is ∣PN∣|PN|?
<spanclass=′latex−bold′>(A)</span> 32<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 32<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> 3 <span class='latex-bold'>(A)</span>\ \dfrac 32 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ \dfrac {\sqrt 3} 2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ \sqrt 3
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