Tile a \(5\times6\) rectangle in an irreducible way by laying \(1\times2\) rectangles.
Robinson found a chest with books and instruments after the ship wreck. Not all the books were in readable condition, but some of the books he managed to read. One sentence read “72 chickens cost *619* p”. (The starred digits were not readable). He has not tasted a chicken for quite some time, and it was pleasant to imagine a properly cooked chicken in front of him. He also was able to decipher the cost of one chicken. Can you?
Jack believes that he can place \(99\) integers in a circle such that for each pair of neighbours the ratio between the larger and smaller number is a prime. Can he be right?
What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular \(n\)-gon?
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).
30 people vote on five proposals. In how many ways can the votes be distributed if everyone votes only for one proposal and only the number of votes cast for each proposal is taken into account?
Prove there are no integer solutions for the equation \(x^2 + 1990 = y^2\).
Solve the equation with natural numbers \(1 + x + x^2 + x^3 = 2y\).
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?
There are three sets of dominoes of different colours. How can you put the dominoes from all three sets into a chain (according to the rules of the game) so that every two neighbouring dominoes are of a different colour?