MathDB

1

Part of 2002 Tournament Of Towns

Problems(5)

Uniqueness question

Source: Tournament of Towns,Spring 2002, Junior O Level, P1

5/13/2014
There are many a×ba\times b rectangular cardboard pieces (a,b∈Na,b\in\mathbb{N} such that a<ba<b). It is given that by putting such pieces together without overlapping one can make 49×5149\times 51 rectangle, and 99×10199\times 101 rectangle. Can one uniquely determine a,ba,b from this?
geometryrectanglenumber theorygreatest common divisorcombinatorics proposedcombinatorics
Easy one

Source: Tournament of Towns,Spring 2002, Junior A Level, P1

5/13/2014
Let a,b,ca,b,c be sides of a triangle. Show that a3+b3+3abc>c3a^3+b^3+3abc>c^3.
inequalitiesinequalities proposedBPSQ
An easy tangent

Source: Tournament of Towns,Spring 2002, Senior A Level, P1

5/14/2014
In a triangle ABCABC it is given tan⁡A,tan⁡B,tan⁡C\tan A,\tan B,\tan C are integers. Find their values.
trigonometrycalculusintegrationgeometry proposedgeometry
Diagonals of a 2002-gon

Source: Tournament of Towns, Fall 2002, Junior O Level, P1

5/15/2014
In a convex 2002-gon2002\text{-gon} several diagonals are drawn so that they do not intersect inside the polygon. As a result the polygon splits into 20002000 triangles. Isit possible that exactly 10001000 triangles have diagonals for all their three sides?
geometry proposedgeometry
Salaries of Employees

Source: Tournament of Towns, Fall 2002, Junior A Level, P1

5/16/2014
There are 20022002 employees in a bank. All the employees came to celebrate the bank's jubilee and were seated around one round table. It is known that the difference in salaries of any two adjacent employees is 22 or 33 dollars. Find the maximal difference in salaries of two employees, if it is known all salaries are different.
combinatorics proposedcombinatorics