Prove that amongst numbers written only using the number 1, i.e.: 1, 11, 111, etc, there is a number than is divisible by 1987.
Prove that there is a power of 3 that ends in 001.
A \(3\times 3\) square is filled with the numbers \(-1, 0, +1\). Prove that two of the 8 sums in all directions – each row, column, and diagonal – will be equal.
100 people are sitting around a round table. More than half of them are men. Prove that there are two males sitting opposite one another.
The numbers \(1, 2, \dots , 9\) are divided into three groups. Prove that the product of the numbers in one of the groups will always be no less than 72.
Some whole numbers are placed into a \(10\times 10\) table, so that the difference between any two neighbouring, horizontally or vertically adjacent, squares is no greater than 5. Prove that there will always be two identical numbers in the table.
Prove that in any group of 6 people there are either three pairs of people who know one another, or three pairs of people who do not know one another.
A warehouse contains 200 boots of each of the sizes 8, 9, and 10. Amongst these 600 boots, 300 are left boots and 300 are right boots. Prove that there are at least 100 usable pairs of boots in the warehouse.
The alphabet of the Ni-Boom-Boom tribe contains 22 consonants and 11 vowels. A word in this language is defined as any combination of letters in which there are no consecutive consonants and no letter is used more than once. The alphabet is divided into 6 non-empty groups. Prove that it is possible to construct a word from all the letters in the group in at least one of the groups.
Prove that in any group of 10 whole numbers there will be a few whose sum is divisible by 10.