MathDB

3

Part of 2008 Bundeswettbewerb Mathematik

Problems(2)

Angle bisector, median and foot of altitude

Source: Germany Bundeswettbewerb Mathematik 2008, Round 1, Problem 3

7/19/2008
Prove: In an acute triangle ABC ABC angle bisector wα, w_{\alpha}, median sb s_b and the altitude hc h_c intersect in one point if wα, w_{\alpha}, side BC BC and the circle around foot of the altitude hc h_c have vertex A A as a common point.
geometryangle bisectorgeometry unsolved
Dissect surface of sphere in eight curvilinear triangles

Source: Germany Bundeswettbewerb Mathematik 2008, Round 2, Problem 3

9/7/2008
Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.
geometry3D geometryspheregeometric transformationreflectioncombinatorics unsolvedcombinatorics