At the end of the month 5 workers were paid a total of £1,500 between them. Each wants to buy themselves a smartphone that costs £320. Prove that one of them will have to wait another month in order to do so.
The total age of a group of 7 people is 332 years. Prove that it is possible to choose three members of this group so that the sum of their ages is no less than 142 years.
Prove that within a group of \(51\) whole numbers there will be two whose difference of squares is divisible by \(100\).
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.