We solve problems

Problem #PRU-21977

Problems Methods Pigeonhole principle Pigeonhole principle (other)

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) Algebra and arithmetic 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 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 Geometric 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 Geometric 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.

Problem #PRU-21984

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

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.

Problem #PRU-21985

Problems Methods Algebraic methods Counting in two ways Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

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.

Problem #PRU-21987

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Prove that within a group of \(51\) whole numbers there will be two whose difference of squares is divisible by \(100\).