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Macedonia North Juniors 2020-21 3p

geometry problems from North Macedonian  Junior Mathematical Olympiad with aops links in the names


2020 - 2021


Let ABC be an isosceles triangle with base AC. Points D and E are chosen on the sides AC and BC, respectively, such that CD = DE. Let H, J, and K be the midpoints of DE, AE, and BD, respectively. The circumcircle of triangle DHK intersects AD at point F, whereas the circumcircle of triangle HEJ intersects BE at G. The line through K parallel to AC intersects AB at I. Let IH \cap GF = {M}. Prove that J, M, and K are collinear points.

Let ABCD be a tangential quadrilateral with inscribed circle k(O,r) which is tangent to the sides BC and AD at K and L, respectively. Show that the circle with diameter OC passes through the intersection point of KL and OD.

Let ABC be an acute triangle and let X and Y be points on the segments AB and AC such that BX = CY. If I_{B} and I_{C} are centers of inscribed circles in triangles ABY and ACX, and T is the second intersection point of the circumcircles of ABY and ACX, show that: \frac{TI_{B}}{TI_{C}} = \frac{BY}{CX}. 


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