We solve problems

Problem #PRU-21980

Problems Methods Pigeonhole principle Pigeonhole principle (other)

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-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-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 \(100\).

Problem #PRU-21990

Problems Fun Problems Word Problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

A \(3\times 3\) square is filled with the numbers \(-1, 0, +1\). Prove that two of the 8 sums in all directions – each row, column, and diagonal – will be equal.

Problem #PRU-21991

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

11–13 2.0

100 people are sitting around a round table. More than half of them are men. Prove that there are two males sitting opposite one another.

Problem #PRU-21995

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 3.0

Prove that in any group of 6 people there are either three pairs of people who know one another, or three pairs of people who do not know one another.

Problem #PRU-21997

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 3.0

A warehouse contains 200 boots of each of the sizes 8, 9, and 10. Amongst these 600 boots, 300 are left boots and 300 are right boots. Prove that there are at least 100 usable pairs of boots in the warehouse.

Problem #PRU-21998

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

The alphabet of the Ni-Boom-Boom tribe contains 22 consonants and 11 vowels. A word in this language is defined as any combination of letters in which there are no consecutive consonants and no letter is used more than once. The alphabet is divided into 6 non-empty groups. Prove that it is possible to construct a word from all the letters in the group in at least one of the groups.