Problems

A system of points connected by segments is called “connected” if from each point one can go to any other one along these segments. Is it possible to connect five points to a connected system so that when erasing any segment, exactly two connected points systems are formed that are not related to each other? (We assume that in the intersection of the segments, the transition from one of them to another is impossible).

Every integer from 1 to 64 is written in an $8\times 8$ table. Prove that no matter the order these numbers are written, there are two adjacent numbers which are at least $5$ apart. (Numbers in cells which share a side are called adjacent).

What is the largest amount of numbers that can be selected from the set 1, 2, ..., 1963 so that the sum of any two numbers is not divisible by their difference?

All of the integers from 1 to 81 are written in a $9\times 9$ table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 6. (Numbers that are in cells which share a common side are called adjacent.)

Prove that in a group of 11 arbitrary infinitely long decimal numbers, it is possible to choose two whose difference contains either, in decimal form, an infinite number of zeroes or an infinite number of nines.

A group of $2n$ people were gathered together. Each person knew at least $n$ 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. ($n\ge 2$)

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.

Several pieces of carpet are laid along a corridor. Pieces cover the entire corridor from end to end without omissions and even overlap one another, so that over some parts of the floor lie several layers of carpet. Prove that you can remove a few pieces, perhaps by taking them out from under others and leaving the rest exactly in the same places they used to be, so that the corridor will still be completely covered and the total length of the pieces left will be less than twice the length corridor.

All integers from 1 to $2n$ are written in a row. Then, to each number, the number of its place in the row is added, that is, to the first number 1 is added, to the second – 2, and so on.

Prove that among the sums obtained there are at least two that give the same remainder when divided by $2n$.