Πέμπτη, 15 Δεκεμβρίου 2016

Baltic Way 2016 - Τέσσερα προβλήματα Γεωμετρίας

16. In triangle ABC, the points D and E are the intersections of the angular bisectors from C and B with the sides AB and AC, respectively. Points F and G on the extensions of AB and AC beyond B and C, respectively, satisfy BF = CG = BC. Prove that F G k DE. 
17. Let ABCD be a convex quadrilateral with AB = AD. Let T be a point on the diagonal AC such that ∠ABT + ∠ADT = ∠BCD. Prove that AT + AC ≥ AB + AD. 18. Let ABCD be a parallelogram such that ∠BAD = 60◦ . Let K and L be the midpoints of BC and CD, respectively. Assuming that ABKL is a cyclic quadrilateral, find ∠ABD. 

19. Consider triangles in the plane where each vertex has integer coordinates. Such a triangle can be legally transformed by moving one vertex parallel to the opposite side to a different point with integer coordinates. Show that if two triangles have the same area, then there exists a series of legal transformations that transforms one to the other. 

20. Let ABCD be a cyclic quadrilateral with AB and CD not parallel. Let M be the midpoint of CD. Let P be a point inside ABCD such that P A = P B = CM. Prove that AB, CD and the perpendicular bisector of MP are concurrent.

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