MathDB

2

Part of 2008 Sharygin Geometry Olympiad

Problems(4)

Right triangle and ratio

Source: Sharygin contest. The final raund. 2008. Grade 8. First day. Problem 2

8/31/2008
(F.Nilov) Given right triangle ABC ABC with hypothenuse AC AC and \angle A \equal{} 50^{\circ}. Points K K and L L on the cathetus BC BC are such that \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}. Determine the ratio CK/LB CK/LB.
ratiotrigonometrygeometrycircumcirclegeometry unsolved
Pedal quadrilaterl with perpendicular diagonals

Source: Sharygin contest. The final raund. 2008. Grade 9. First day. Problem 2

8/31/2008
(F.Nilov) Given quadrilateral ABCD ABCD. Find the locus of points such that their projections to the lines AB AB, BC BC, CD CD, DA DA form a quadrilateral with perpendicular diagonals.
geometry unsolvedgeometry
Construction with medial triangle

Source: Sharygin contest. The final raund. 2008. Grade 10. First day. Problem 2

8/31/2008
(A.Myakishev) Let triangle A1B1C1 A_1B_1C_1 be symmetric to ABC ABC wrt the incenter of its medial triangle. Prove that the orthocenter of A1B1C1 A_1B_1C_1 coincides with the circumcenter of the triangle formed by the excenters of ABC ABC.
geometryincentercircumcircleparallelogramgeometric transformationreflectionratio
Construct two concentric circles

Source: Sharygin contest 2008. The correspondence round. Problem 2

9/3/2008
(V.Protasov, 8) For a given pair of circles, construct two concentric circles such that both are tangent to the given two. What is the number of solutions, depending on location of the circles?
geometrygeometric transformationreflectiongeometry proposed