You are given \(7\) straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than \(26^{\circ}\).
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).
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, 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.
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 \(2n\).
In draughts, the king attacks by jumping over another draughts-piece. What is the maximum number of draughts kings we can place on the black squares of a standard \(8\times 8\) draughts board, so that each king is attacking at least one other?
What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?