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).
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.
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 \(2n\).
What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?
The numbers \(1, 2, 3, \dots , 99\) are written onto 99 blank cards in order. The cards are then shuffled and then spread in a row face down. The numbers \(1, 2, 3, \dots, 99\) are once more written onto in the blank side of the cards in order. For each card the numbers written on it are then added together. The 99 resulting summations are then multiplied together. Prove that the result will be an even number.
The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.
The numbers \(a_1, a_2, \dots , a_{1985}\) are the numbers \(1, 2, \dots , 1985\) rearranged in some order. Each number \(a_k\) is multiplied by its number \(k\), and then the largest number is chosen among the resulting 1985 products. Prove that it is not less than \(993^2\).
The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.