4
Part of 1990 Bundeswettbewerb Mathematik
Problems(2)
midpoints of the six edges of a tetrahedron lie on a sphere
Source: Germany Federal - Bundeswettbewerb Mathematik 1990, round 1, p4
2/21/2020
Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.
spheremidpoints3D geometrytetrahedrongeometry
worm in the plane
Source: Bundeswettbewerb Mathematik 1990, stage 2, problem 4
6/18/2004
In the plane there is a worm of length 1. Prove that it can be always covered by means of half of a circular disk of diameter 1.
Note. Under a "worm", we understand a continuous curve. The "half of a circular disk" is a semicircle including its boundary.
inequalitiesfunctiongeometry3D geometryspheregeometry solved