MathDB

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Part of 2011 Sharygin Geometry Olympiad

Problems(4)

Dividing a convex heptagon

Source: 2011 Sharygin Geometry Olympiad #1

5/22/2014
Does a convex heptagon exist which can be divided into 2011 equal triangles?
geometryrectanglegeometry unsolved
isosceles criterion for a trapezoid with perpendicular diagonals

Source: Sharygin 2011 Final 8.1

12/13/2018
The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles
geometrytrapezoiddiagonalsperpendicularisosceles
tangent circumcircles starting with a semicircle and two altitudes

Source: Sharygin 2011 Final 9.1

12/16/2018
Altitudes AA1AA_1 and BB1BB_1 of triangle ABC meet in point HH. Line CHCH meets the semicircle with diameter ABAB, passing through A1,B1A_1, B_1, in point DD. Segments ADAD and BB1BB_1 meet in point MM, segments BDBD and AA1AA_1 meet in point NN. Prove that the circumcircles of triangles B1DMB_1DM and A1DNA_1DN touch.
geometrycircumcircletangent circlesaltitudessemicircle
tangent circles when centroid lies on circle of midpoints of AC,AB and C

Source: Sharygin 2011 Final 10.1

3/31/2019
In triangle ABCABC the midpoints of sides AC,BCAC, BC, vertex CC and the centroid lie on the same circle. Prove that this circle touches the circle passing through A,BA, B and the orthocenter of triangle ABCABC.
geometryorthocenterCentroidtangent circles