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find angle and ratios of angles in acute triangle

Source: Albanian IMO 2011 TST

May 27, 2011
ratioconicshyperbolatrigonometryanalytic geometrycircumcirclegeometry unsolved

Problem Statement

In the acute angle triangle ABCABC the point OO is the center of the circumscribed circle and the lines OA,OB,OCOA,OB,OC intersect sides BC,CA,ABBC,CA,AB respectively in points M,N,PM,N,P such that ∠NMP=90o\angle NMP=90^o.
(a) Find the ratios ∠AMN∠NMC\frac{\angle AMN}{\angle NMC},∠AMP∠PMB\frac{\angle AMP}{\angle PMB}.
(b) If any of the angles of the triangle ABCABC is 60o60^o, find the two other angles.
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