All of the integers from 1 to 64 are written in an \(8 \times 8\) table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 5. (Numbers that are in cells which share a common side are called adjacent).
A group of \(2n\) people were gathered together, of whom each person knew no less than \(n\) of the other people present. Prove that it is possible to select 4 people and seat them around a table so that each person sits next to people they know. (\(n \geq 2\))
Two players play on a square field of size \(99 \times 99\), which has been split onto cells of size \(1 \times 1\). The first player places a cross on the center of the field; After this, the second player can place a zero on any of the eight cells surrounding the cross of the first player. After that, the first puts a cross onto any cell of the field next to one of those already occupied, etc. The first player wins if he can put a cross on any corner cell. Prove that with any strategy of the second player the first can always win.
30 teams are taking part in a football championship. Prove that at any moment in the contest there will be two teams who have played the same number of matches up to that moment, assuming every team plays every other team exactly once by the end of the tournament.
What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?
Is it possible to place the numbers \(1, 2,\dots 12\) around a circle so that the difference between any two adjacent numbers is 3, 4, or 5?
Airlines connect pairs of cities. How can you connect 50 cities with the fewest number of airlines so that from every city you can get to any other city by taking at most two flights?
In a corridor of length 100 m, 20 sections of red carpet are laid out. The combined length of the sections is 1000 m. What is the largest number there can be of distinct stretches of the corridor that are not covered by carpet, given that the sections of carpet are all the same width as the corridor?
Two lines on the plane intersect at an angle \(\alpha\). On one of them there is a flea. Every second it jumps from one line to the other (the point of intersection is considered to belong to both straight lines). It is known that the length of each of her jumps is 1 and that she never returns to the place where she was a second ago. After some time, the flea returned to its original point. Prove that for the angle \(\alpha\) the value \(\alpha/\pi\) is a rational number.
The White Rook pursues a black horse on a board of \(3 \times 1969\) cells (they walk in turn according to the usual rules). How should the rook play in order to take the horse? White makes the first move.