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19th kmo #2

Source: KMO round 2, problem 2

February 3, 2006
geometrycircumcirclegeometry unsolved

Problem Statement

For triangle ABCABC, PP and QQ satisfy ∠BPA+∠AQC=90∘\angle BPA + \angle AQC=90^{\circ}. It is provided that the vertices of the triangle BAPBAP and ACQACQ are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be NN (A≠NA \neq N, however if AA is the only intersection A=NA=N), and the midpoint of segment BCBC be MM. Show that the length of MNMN does not depend on PP and QQ.
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