MathDB

1

Part of 2009 Tournament Of Towns

Problems(6)

Cut a square - TT 2009 Junior-O1

Source:

9/3/2010
Is it possible to cut a square into nine squares and colour one of them white, three of them grey and ve of them black, such that squares of the same colour have the same size and squares of different colours will have different sizes?
(3 points)
Pirates - TT 2009 Senior-A1

Source:

9/3/2010
One hundred pirates played cards. When the game was over, each pirate calculated the amount he won or lost. The pirates have a gold sand as a currency; each has enough to pay his debt.
Gold could only change hands in the following way. Either one pirate pays an equal amount to every other pirate, or one pirate receives the same amount from every other pirate.
Prove that after several such steps, it is possible for each winner to receive exactly what he has won and for each loser to pay exactly what he has lost.
(4 points)
induction
At most 10 moves - TT 2009 Junior-A1

Source:

9/3/2010
Each of 1010 identical jars contains some milk, up to 1010 percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other 99 jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most 1010 moves.
(4 points)
percentcombinatorics unsolvedcombinatorics
2009 ToT Spring Junior A P1 2 numbers on board, 1/2009, 1/2008

Source:

3/7/2020
There are two numbers on a board, 1/20091/2009 and 1/20081/2008. Alex and Ben play the following game. At each move, Alex names a number xx (of his choice), while Ben responds by increasing one of the numbers on the board (of his choice) by xx. Alex wins if at some moment one of the numbers on the board becomes 11. Can Alex win (no matter how Ben plays)?
combinatoricsgame
2009 ToT Spring Junior O P1 diagonals in a convex 2009-gon

Source:

3/7/2020
In a convex 20092009-gon, all diagonals are drawn. A line intersects the 20092009-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.
convex polygonconvexdiagonalcombinatorial geometryEvengeometry
2009 ToT Spring Senior A P1 rectangle dissected into rectangles

Source:

3/7/2020
A rectangle is dissected into several smaller rectangles. Is it possible that for each pair of these rectangles, the line segment connecting their centers intersects some third rectangle?
combinatorial geometrygeometryrectangle