MathDB

5

Part of 2008 Sharygin Geometry Olympiad

Problems(4)

are two triangles similar?

Source: Sharygin contest. The final raund. 2008. Grade 8. First day. Problem 5

8/31/2008
(A.Zaslavsky) Given two triangles ABC ABC, A′B′C′ A'B'C'. Denote by α \alpha the angle between the altitude and the median from vertex A A of triangle ABC ABC. Angles β \beta, γ \gamma, α′ \alpha', β′ \beta', γ′ \gamma' are defined similarly. It is known that \alpha \equal{} \alpha', \beta \equal{} \beta', \gamma \equal{} \gamma'. Can we conclude that the triangles are similar?
geometry unsolvedgeometry
tetrahedron and hexagons

Source: Sharygin contest. The final raund. 2008. Grade 9. First day. Problem 5

8/31/2008
(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?
geometry3D geometrytetrahedrongeometry unsolved
Section of a regular tetragonal pyramid

Source: Sharygin contest. The final raund. 2008. Grade 10. First day. Problem 5

8/31/2008
(I.Bogdanov) A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid.
geometry3D geometrypyramidratiogeometry unsolved
Reconstruct the square

Source: Sharygin contest 2008. The correspondence round. Problem 5

9/3/2008
(Kiev olympiad, 8--9) Reconstruct the square ABCD ABCD, given its vertex A A and distances of vertices B B and D D from a fixed point O O in the plane.
geometrygeometric transformationrotationgeometry proposed