MathDB

6

Part of 2009 Tournament Of Towns

Problems(3)

Infinite chessboard - TT 2009 Junior-A6

Source:

9/3/2010
On an in finite chessboard are placed 2009 n×n2009 \ n \times n cardboard pieces such that each of them covers exactly n2n^2 cells of the chessboard. Prove that the number of cells of the chessboard which are covered by odd numbers of cardboard pieces is at least n2.n^2.
(9 points)
modular arithmetic
2009 ToT Spring Junior A P7 sum of triangle areas

Source:

2/26/2020
Angle CC of an isosceles triangle ABCABC equals 120o120^o. Each of two rays emitting from vertex CC (inwards the triangle) meets ABAB at some point (PiP_i) reflects according to the rule the angle of incidence equals the angle of reflection" and meets lateral side of triangle ABCABC at point QiQ_i (i=1,2i = 1,2). Given that angle between the rays equals 60o60^o, prove that area of triangle P1CP2P_1CP_2 equals the sum of areas of triangles AQ1P1AQ_1P_1 and BQ2P2BQ_2P_2 (AP1<AP2AP_1 < AP_2).
isoscelesgeometryareas
2009 ToT Spring Senior A P6 marking points in circle game

Source:

3/7/2020
An integer n>1n > 1 is given. Two players in turns mark points on a circle. First Player uses red color while Second Player uses blue color. The game is over when each player marks nn points. Then each player nds the arc of maximal length with ends of his color, which does not contain any other marked points. A player wins if his arc is longer (if the lengths are equal, or both players have no such arcs, the game ends in a draw). Which player has a winning strategy?
combinatorial geometrygame strategy