You are given 8 different natural numbers that are no greater than 15. Prove that there are three pairs of these numbers whose positive difference is the same.
Prove that in any group of 5 people there will be two who know the same number of people in that group.
Several football teams are taking part in a football tournament, where each team plays every other team exactly once. Prove that at any point in the tournament there will be two teams who have played exactly the same number of matches up to that point.
a) What is the maximum number of squares on an \(8\times 8\) grid that can be shaded in with a black pen such that each ‘L’ shaped group of 3 squares has at least one unshaded square.
b) What is the maximum number of squares on an \(8\times 8\) grid that can be shaded in with a black pen, such that each ‘L’ shaped group of 3 squares has at least one shaded square.
Prove that it is not possible to completely cover an equilateral triangle with two smaller equilateral triangles.
51 points were thrown into a square of side 1 m. Prove that it is possible to cover some set of 3 points with a square of side 20 cm.
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.