MathDB

6

Part of 2008 Sharygin Geometry Olympiad

Problems(4)

Acute-angled or obtuse-angled triangles in regular 2008-gon

Source: Sharygin contest. The final raund. 2008. Grade 8. Second day. Problem 6

8/31/2008
(B.Frenkin) Consider the triangles such that all their vertices are vertices of a given regular 2008-gon. What triangles are more numerous among them: acute-angled or obtuse-angled?
geometry unsolvedgeometry
Construct the triangle

Source: Sharygin contest. The final raund. 2008. Grade 9. Second day. Problem 6

8/31/2008
(B.Frenkin) Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.
geometrygeometric transformationreflectioncircumcirclegeometry unsolved
bc=8Rr

Source: Sharygin contest. The final raund. 2008. Grade 10. Second day. Problem 6

8/31/2008
(B.Frenkin) The product of two sides in a triangle is equal to 8Rr 8Rr, where R R and r r are the circumradius and the inradius of the triangle. Prove that the angle between these sides is less than 60āˆ˜ 60^{\circ}.
geometrycircumcircleinradiustrigonometrygeometry unsolved
Determine the locus of points

Source: Sharygin contest 2008. The correspondence round. Problem 6

9/3/2008
(A. Myakishev, 8--9) In the plane, given two concentric circles with the center A A. Let B B be an arbitrary point on some of these circles, and C C on the other one. For every triangle ABC ABC, consider two equal circles mutually tangent at the point K K, such that one of these circles is tangent to the line AB AB at point B B and the other one is tangent to the line AC AC at point C C. Determine the locus of points K K.
geometry proposedgeometry