MathDB

8

Part of 2008 Sharygin Geometry Olympiad

Problems(4)

Can Boris define its perimeter?

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

8/31/2008
(B.Frenkin, A.Zaslavsky) A convex quadrilateral was drawn on the blackboard. Boris marked the centers of four excircles each touching one side of the quadrilateral and the extensions of two adjacent sides. After this, Alexey erased the quadrilateral. Can Boris define its perimeter?
geometryperimetergeometry unsolved
Three lines are concurent

Source: Sharygin contest. The final raund. 2008. Grade 9. Second day. Problem 8

8/31/2008
(J.-L.Ayme, France) Points P P, Q Q lie on the circumcircle ω \omega of triangle ABC ABC. The perpendicular bisector l l to PQ PQ intersects BC BC, CA CA, AB AB in points A′ A', B′ B', C′ C'. Let A" A", B" B", C" C" be the second common points of l l with the circles A′PQ A'PQ, B′PQ B'PQ, C′PQ C'PQ. Prove that AA" AA", BB" BB", CC" CC" concur.
geometrycircumcircle
Sets on plane with (3,2)-property

Source: Sharygin contest. The final raund. 2008. Grade 10. Second day. Problem 8

8/31/2008
(A.Akopyan, V.Dolnikov) Given a set of points inn the plane. It is known that among any three of its points there are two such that the distance between them doesn't exceed 1. Prove that this set can be divided into three parts such that the diameter of each part does not exceed 1.
geometry unsolvedgeometry
convex can be dissected into obtuse triangles

Source: Sharygin contest 2008. The correspondence round. Problem 8

9/3/2008
(T.Golenishcheva-Kutuzova, B.Frenkin, 8--11) a) Prove that for n>4 n > 4, any convex n n-gon can be dissected into n n obtuse triangles.
geometry proposedgeometry