MathDB

5

Part of 2009 Tournament Of Towns

Problems(7)

A new website - TT 2009 Junior-O5

Source:

9/3/2010
A new website registered 20002000 people. Each of them invited 10001000 other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?
(5 points)
2009 ToT Spring Junior O P5 circumcenter lies on rhombus diagonal

Source:

2/26/2020
In rhombus ABCDABCD, angle AA equals 120o120^o. Points MM and NN are chosen on sides BCBC and CDCD so that angle NAMNAM equals 30o30^o. Prove that the circumcenter of triangle NAMNAM lies on a diagonal of of the rhombus.
geometryrhombusCircumcenter
Weights - TT 2009 Junior-A5

Source:

9/3/2010
We have N objects with weights 1,2,⋯ ,N1, 2,\cdots , N grams. We wish to choose two or more of these objects so that the total weight of the chosen objects is equal to average weight of the remaining objects. Prove that
(a) (2 points) if N+1N + 1 is a perfect square, then the task is possible;
(b) (6 points) if the task is possible, then N+1N + 1 is a perfect square.
Country with two capitals - TT 2009 Senior-O5

Source:

9/3/2010
A country has two capitals and several towns. Some of them are connected by roads. Some of the roads are toll roads where a fee is charged for driving along them. It is known that any route from the south capital to the north capital contains at least ten toll roads. Prove that all toll roads can be distributed among ten companies so that anybody driving from the south capital to the north capital must pay each of these companies.
(5 points)
combinatorics unsolvedcombinatorics
Area of the triangle - TT 2009 Senior-A5

Source:

9/3/2010
Let XYZXY Z be a triangle. The convex hexagon ABCDEFABCDEF is such that AB;CDAB; CD and EFEF are parallel and equal to XY;YZXY; Y Z and ZXZX, respectively. Prove that area of triangle with vertices at the midpoints of BC;DEBC; DE and FAFA is no less than area of triangle XYZ.XY Z.
(8 points)
geometry
2009 ToT Spring Junior A P5 knights guard a 9-tower circular wall

Source:

3/7/2020
A castle is surrounded by a circular wall with 99 towers which are guarded by knights during the night. Every hour the castle clock strikes and the guards shift to the neighboring towers, each guard always moves in the same direction (either clockwise or counterclockwise). Given that (i) during the night each knight guards every tower (ii) at some hour each tower was guarded by at least two knights (iii) at some hour exactly 55 towers were guarded by single knights, prove that at some hour one of the towers was unguarded.
combinatorics
2009 ToT Spring Senior O P5 concurrency in a tetrahedron

Source:

2/26/2020
Suppose that XX is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting XX with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.
tetrahedron3D geometrygeometryconcurrent