Problems

There is a $5\times 9$ rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?

The centres of all unit squares are marked in a $10\times 10$ chequered box (100 points in total). What is the smallest number of lines, that are not parallel to the sides of the square, that are needed to be drawn to erase all of the marked points?

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?

A game of ’Battleships’ has a fleet consisting of one $1\times 4$ square, two $1\times 3$ squares, three $1\times 2$ squares, and four $1\times 1$ squares. It is easy to distribute the fleet of ships on a $10\times 10$ board, see the example below. What is the smallest square board on which this fleet can be placed? Note that by the rules of the game, no two ships can be placed on horizontally, vertically, or diagonally adjacent squares.

One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?

The city plan is a rectangle of $5\times 10$ cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?

On a $100\times 100$ board 100 rooks are placed that cannot capturing one another.

Prove that an equal number of rooks is placed in the upper right and lower left cells of $50\times 50$ squares.

An endless board is painted in three colours (each cell is painted in one of the colours). Prove that there are four cells of the same colour, located at the vertices of the rectangle with sides parallel to the side of one cell.

A board of size $2005\times 2005$ is divided into square cells with a side length of 1 unit. Some board cells are numbered in some order by numbers 1, 2, ... so that from any non-numbered cell there is a numbered cell within a distance of less than 10. Prove that there can be found two cells with a distance between them of less than 150, which are numbered by numbers that differ by more than 23. (The distance between the cells is the distance between their centres.)