We solve problems

Problem #PRU-21974

Problems Methods Pigeonhole principle Pigeonhole principle (other)

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.

Problem #PRU-21975

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

In Scotland there are $m$ football teams containing 11 players each. All of the players met at the airport in order to travel to England for a match. The plane made 10 journeys from Scotland to England, carrying 10 passengers each time. One player also flew to the location of the match by helicopter. Prove that at least one team made it in its entirety to the other country to play the match.

Problem #PRU-21976

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

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.

Problem #PRU-21977

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Prove that in any group of 5 people there will be two who know the same number of people in that group.

Problem #PRU-21978

Problems Methods Pigeonhole principle Pigeonhole principle (other) Fun Problems Word Problems Tables and tournaments Tournament tables

11–13 3.0

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.

Problem #PRU-21979

Problems Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle

12–13 3.0

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.

Problem #PRU-21980

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–12 3.0

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.

Problem #PRU-21981

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–12 2.0

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.

Problem #PRU-21982

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

11–13 3.0

Prove that it is not possible to completely cover an equilateral triangle with two smaller equilateral triangles.

Problem #PRU-21983

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

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.