MathDB

6

Part of 2003 Tournament Of Towns

Problems(4)

Prove that the angle AED is not acute

Source: Tournament of Towns Spring 2003 - Junior A-Level - Problem 6

6/14/2011
A trapezoid with bases ADAD and BCBC is circumscribed about a circle, EE is the intersection point of the diagonals. Prove that ∠AED\angle AED is not acute.
geometrytrapezoidinequalitiesgeometric transformationreflectiontriangle inequalitygeometry unsolved
Placing + and - signs in a 4 × 4 square

Source: Tournament of Towns Spring 2003 - Senior A-Level - Problem 6

6/15/2011
The signs "++" or "−-" are placed in all cells of a 4×44 \times 4 square table. It is allowed to change a sign of any cell altogether with signs of all its adjacent cells (i.e. cells having a common side with it). Find the number of different tables that could be obtained by iterating this procedure.
combinatorics unsolvedcombinatorics
Most distant point from vertex on rectangular parallelepiped

Source: ToT 2003 JA-6, SA-3.

6/26/2011
An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?
geometry unsolvedgeometry
Proving that five points are coplanar

Source: ToT 2003 SA-6

7/3/2011
Let OO be the center of insphere of a tetrahedron ABCDABCD. The sum of areas of faces ABCABC and ABDABD equals the sum of areas of faces CDACDA and CDBCDB. Prove that OO and midpoints of BC,AD,ACBC, AD, AC and BDBD belong to the same plane.
geometry3D geometrytetrahedronparallelograminradiusgeometry unsolved