3
Part of 1990 Vietnam National Olympiad
Problems(2)
Vietnam NMO 1990_3
Source:
10/26/2008
A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron.
(a) Can this be done so that the volume of the parallelepiped is at least of the volume of the tetrahedron?
(b) Determine the common point of the three planes if the volume of the parallelepiped is of the volume of the tetrahedron.
geometry3D geometrytetrahedrongeometry unsolved
Vietnam NMO 1990_6
Source:
10/26/2008
The children sitting around a circle are playing the game as follows. At first the teacher gives each child an even number of candies (bigger than , may be equal, maybe different). A certain child gives half of his candies to his neighbor on the right. Then the child who has just received candies does the same if he has an even number of candies, otherwise he gets one candy from the teacher and then does the job; and so on. Prove that after several steps there will be a child who will be able, giving the teacher half of his candies, to make the numbers of candies of all the children equal.
combinatorics unsolvedcombinatorics