Every point in a plane, which has whole-number co-ordinates, is plotted in one of \(n\) colours. Prove that there will be a rectangle made out of 4 points of the same colour.
\(2n\) diplomats sit around a round table. After a break the same \(2n\) diplomats sit around the same table, but this time in a different order.
Prove that there will always be two diplomats with the same number of people sitting between them, both before and after the break.
A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.
On every cell of a \(9 \times 9\) board there is a beetle. At the sound of a whistle, every beetle crawls onto one of the diagonally neighbouring cells. Note that, in some cells, there may be more than one beetle, and some cells will be unoccupied.
Prove that there will be at least 9 unoccupied cells.
Prove that if \(a, b, c\) are odd numbers, then at least one of the numbers \(ab-1\), \(bc-1\), \(ca-1\) is divisible by 4.
10 natural numbers are written on a blackboard. Prove that it is always possible to choose some of these numbers and write “\(+\)” or “\(-\)” between them so that the resulting algebraic sum is divisible by 1001.
Is it possible to fill an \(n\times n\) table with the numbers \(-1\), \(0\), \(1\), such that the sums of all the rows, columns, and diagonals are unique?
Upon the installation of a keypad lock, each of the 26 letters located on the lock’s keypad is assigned an arbitrary natural number known only to the owner of the lock. Different letters do not necessarily have different numbers assigned to them. After a combination of different letters, where each letter is typed once at most, is entered into the lock a summation is carried out of the corresponding numbers to the letters typed in. The lock opens only if the result of the summation is divisible by 26. Prove that for any set of numbers assigned to the 26 letters, there exists a combination that will open the lock.
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.
A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?