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Brazilian Girls in Mathematics Tournament 2019-21 (TM^2) 3p

geometry problems from Brazilian Girls in Mathematics Tournament (TM^2 ) with aops links in the names

Torneio Meninas na Matematica

started in 2019

2019, 2021
it didn't take place in 2020


Let ABC be a right triangle with hypotenuse BC and center I. Let bisectors of the angles \angle B and \angle C intersect the sides AC and AB in D and E, respectively. Let P and Q be the feet of the perpendiculars of the points D and E on the side BC. Prove that I is the circumcenter of APQ.

2019 Brazilian Girls in Mathematics Tournament p5
Let ABC be an isosceles triangle with AB = AC. Let X and K points over AC and AB, respectively, such that KX = CX. Bisector of \angle AKX intersects line BC at Z. Show that XZ passes through the midpoint of BK.

Let \vartriangle ABC be a triangle in which \angle ACB = 40^o and \angle BAC = 60^o . Let D be a point inside the segment BC such that CD =\frac{AB}{2} and let M be the midpoint of the segment AC. How much is the angle \angle CMD in degrees?


source: www.tm2.org.br

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