Problems

The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

Cutting into four parts. Cut each of the figures below into four equal parts (you can cut along the sides and diagonals of cells).

Prove that the number of all arrangements of the largest possible amount of peaceful bishops (figures that move on diagonals and don’t threaten each other) on the \(8\times 8\) chessboard is an exact square.

Giuseppe has a sheet of plywood, measuring \(22 \times 15\). Giuseppe wants to cut out as many rectangular blocks of size \(3 \times 5\) as possible. How should he do it?

Fred and George had two square cakes. Each twin made two straight cuts on his cake from edge to edge. However, one ended up with three pieces, and the other with four. How could this be?

Deep in a forest there is a small town of talking animals. Elephant, Crocodile, Rabbit, Monkey, Bear, Heron and Fox are friends. They each have a landline telephone and each two telephones are connected by a wire. How many wires were required?

Cut a square into three pieces, from which you can construct a triangle with three acute angles and three different sides.

Cut the board shown in the figure into four congruent parts so that each of them contains three shaded cells. Where the shaded cells are placed in each part need not be the same.

Cut the figure (on the boundaries of cells) into three equal parts (the same in shape and size).

A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?