An infinite sequence of digits is given. Prove that for any natural number \(n\) that is relatively prime with a number 10, you can choose a group of consecutive digits, which when written as a sequence of digits, gives a resulting number written by these digits which is divisible by \(n\).
Prove that, for any integer \(n\), among the numbers \(n, n + 1, n + 2, \dots , n + 9\) there is at least one number that is mutually prime with the other nine numbers.
If we are given any 100 whole numbers then amongst them it is always possible to choose one, or several of them, so that their sum gives a number divisible by 100. Prove that this is the case.
Find the locus of points whose coordinates \((x, y)\) satisfy the relation \(\sin(x + y) = 0\).
Note that if you turn over a sheet on which numbers are written, then the digits 0, 1, 8 will not change and the digits 6 and 9 will switch places, whilst the others will lose their meaning. How many nine-digit numbers exist that do not change when a sheet is turned over?
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).
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.
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 \(2n\).
What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?