BRATISLAVA MATHEMATICS CORRESPONDENCE SEMINAR
Faculty of Mathematics and Physics of Comenius University
Union of Slovak Mathematicians and Physicists
Centre of Spare Time - IUVENTA

BMCS - Problem set of the 3rd spring series 1996/97

Stereometry

  1. Let's divide a space into three subsets with empty intersections. Prove that at least one of them (X) has the following property: For each a>0 there exist two points A, B in X such that |AB|=a.

  2. In the tetrahedron ABCD, the edges AB and CD are perpendicular. BC and AD are perpendicular too. Prove that AC and BD are perpendicular.

  3. All three edges of the base of a pyramid are tangets of a sphere. The sphere touches the base edges in their centers and intersects non-base edges in their centers. Prove that the pyramid is regular (the base is an equilateral triangle and three non-base edges have the same lengths).

  4. Find the lengths of all solid diagonals of polyhedron with 12 sides (dodecahedron) if length of its edge is 1.

  5. There are two vertices of the tetrahedron ABCD on the sphere with radius 10. The remaining two vertices are on the sphere with radius 2. Both spheres have the same center. What is the maximum volume of tetrahedron ABCD?