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