MathDB

6

Part of 2001 Tournament Of Towns

Problems(4)

Triangle Proof

Source: ToT - 2001 Spring Junior A-Level #6

8/17/2011
Let AHAAH_A, BHBBH_B and CHCCH_C be the altitudes of triangle △ABC\triangle ABC. Prove that the triangle whose vertices are the intersection points of the altitudes of triangles △AHBHC\triangle AH_BH_C, △BHAHC\triangle BH_AH_C and △CHAHB\triangle CH_AH_B is equal to triangle △HAHBHC\triangle H_AH_BH_C.
geometrygeometry unsolved
Convex Polyhedra

Source: ToT - 2001 Spring Senior A-Level #6

8/17/2011
Prove that there exist 20012001 convex polyhedra such that any three of them do not have any common points but any two of them touch each other (i.e., have at least one common boundary point but no common inner points).
geometry unsolvedgeometry
Numbers in a Row

Source: ToT - 2001 Fall Junior A-Level #6

8/17/2011
Several numbers are written in a row. In each move, Robert chooses any two adjacent numbers in which the one on the left is greater than the one on the right, doubles each of them and then switches them around. Prove that Robert can make only a finite number of moves.
inductioncombinatorics unsolvedcombinatorics
23 Boxes

Source: ToT - 2001 Fall Senior A-Level #6

8/17/2011
In a row are 23 boxes such that for 1≤k≤231\le k \le 23, there is a box containing exactly kk balls. In one move, we can double the number of balls in any box by taking balls from another box which has more. Is it always possible to end up with exactly kk balls in the kk-th box for 1≤k≤231\le k\le 23?
inductioncombinatorics unsolvedcombinatorics