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SAFEST Olympiad 2019-21 (S.Africa - Estonia) 3p

geometry problems from SAFEST Olympiad, by South African - Estonian IMO teams
with aops links in the names

not in Shortlist


2019-21

2019 SAFEST olympiad p1
Let ABC be an isosceles triangle with AB = AC. Let AD be the diameter of the circumcircle of ABC and let P be a point on the smaller arc BD. The line DP intersects the rays AB and AC at points M and N, respectively. The line AD intersects the lines BP and CP at points Q and R, respectively. Prove that the midpoint of MN lies on the circumcircle of PQR

Let O be the circumcenter and H the orthocenter of an acute-triangle ABC. The perpendicular bisector of AO intersects  the line BC at point S. Let L be the midpoint of OH. Prove that \angle OAH = \angle  LSA.

Let ABC be a triangle with AB > AC. Let D be a point on the side AB such that DB = DC and let M be the midpoint of AC. The line parallel to BC passing through D intersects the line BM in K. Show that \angle KCD = \angle DAC.


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