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Part of 2020 Ukrainian Geometry Olympiad - December

Problems(4)

<E wanted in pentagon ABCDE, AB BC=CD=DE, <C = <D =108^o, <B = 96^o

Source: December 2020 Ukraine Geometry Olympiad VIII p3

12/21/2020
About the pentagon ABCDEABCDE we know that AB=BC=CD=DEAB = BC = CD = DE, ∠C=∠D=108o\angle C = \angle D =108^o, ∠B=96o\angle B = 96^o. Find the value in degrees of ∠E\angle E.
geometrypentagonequal segmentsangles
no of triangles by 1020 points inside a 1000-gon

Source: December 2020 Ukraine Geometry Olympiad IX p3

12/21/2020
Given convex 10001000-gon. Inside this polygon, 10201020 points are chosen so that no 33 of the 20202020 points do not lie on one line. Polygon is cut into triangles so that these triangles have vertices only those specified 20202020 points and each of these points is the vertex of at least one of cutting triangles. How many such triangles were formed?
geometrycombinatoricscombinatorial geometry
tan < DCB wanted, when median creates isosceles triangle, <CAB =30^o

Source: December 2020 Ukraine Geometry Olympiad X p3

12/20/2020
In a triangle ABCABC with an angle ∠CAB=30o\angle CAB =30^o draw median CDCD. If the formed △ACD\vartriangle ACD is isosceles, find tan ∠DCB\angle DCB.
anglesisoscelesmediangeometry
ratio BC/FG, 2 circumcircles related

Source: December 2020 Ukraine Geometry Olympiad XI p3

12/20/2020
On the sides ABAB and ACAC of a triangle ABCABC select points DD and EE respectively, such that AB=6AB = 6, AC=9AC = 9, AD=4AD = 4 and AE=6AE = 6. It is known that the circumscribed circle of △ADE\vartriangle ADE interects the side BCBC at points F,GF, G , where BF<BGBF < BG. Knowing that the point of intersection of lines DFDF and EGEG lies on the circumscribed circle of △ABC\vartriangle ABC , find the ratio BC:FGBC:FG.
circumcirclegeometryratio