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Part of 2011 Albania Team Selection Test

Problems(1)

find angle and ratios of angles in acute triangle

Source: Albanian IMO 2011 TST

5/27/2011
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.
ratioconicshyperbolatrigonometryanalytic geometrycircumcirclegeometry unsolved