A forest contains a million fir trees. It is known that any given tree has at most 600,000 needles. Prove that there will be two trees with the same number of needles.
You are given 12 different whole numbers. Prove that it is possible to choose two of these whose difference is divisible by 11.
A supermarket received a delivery of 25 crates of apples of 3 different types; each crate contains only one type of apple. Prove that there are at least 9 crates of apples of the same sort in the delivery.
In Scotland there are
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
b) What is the maximum number of squares on an
10 school students took part in a Mathematical Olympiad and solved 35 problems in total. It is known that there were students who solved exactly one problem, students who solved exactly two problems, and students who solved exactly three problems. Prove that there is a student who solved exactly 5 problems.
What is the maximum number of kings you could place on a chess board such that no two of them were attacking each other – that is, no two kings are on horizontally, vertically, or diagonally adjacent squares. Kings can move in any direction, but only one square at a time.