Problems

Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).

A square grid on the plane and a triangle with vertices at the nodes of the grid are given. Prove that the tangent of any angle in the triangle is a rational number.

Prove that there is at most one point of an integer lattice on a circle with centre at \((\sqrt 2, \sqrt 3)\).

Does there exist an irreducible tiling with \(1\times2\) rectangles of a \(4\times 6\) rectangle?

Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(5\times8\) rectangle.

Tile the whole plane with the following shapes:

David Smith cut out 12 nets. He claimed that it was possible to make a cube out of each net. Roger Penrose looked at the patterns, and after some considerable thought decided that he was able to make cubes from all the nets except one. Can you figure out which net cannot make a cube?

It is known that it is possible to cover the plane with any cube’s net. Show how you can cover the plane with this net:

In how many ways can you rearrange the numbers 1, 2, ..., 100 so the neighbouring numbers differ by not more than 1?

A strange wonderland creature is called a painting chameleon. If the queen puts the painting chameleon on a chess-like board then he moves one square at a time along the board either horizontally or vertically. When he moves, he either changes his colour to the colour of the square he moves to, or he paints the square he moves to into his own colour. The queen puts a white painting chameleon on an all-black board \(8\times8\) and orders the chameleon to paint the board into a chessboard. Can he succeed?