We solve problems

a) The sports club has 30 members, of which four people are required to participate in the 1,000 metre race. How many ways can this be done?

b) How many ways can I build a team of four people to participate in the relay race 100 m + 200 m + 300 m + 400 m?

Problem #PRU-30737

Problems Combinatorics

11–12 2.0

How many ways can you build a closed line whose vertices are the vertices of a regular hexagon (the line can be self-intersecting)?

Problem #PRU-30748

Problems Combinatorics Integer lattices Polygons and polyhedra with vertices in lattice points

12–13 2.0

Find the number of rectangles made up of the cells of a board with \(m\) horizontals and \(n\) verticals that contain a cell with the coordinates \((p, q)\).

Problem #PRU-30815

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

Each of the edges of a complete graph consisting of 6 vertices is coloured in one of two colours. Prove that there are three vertices, such that all the edges connecting them are the same colour.

Problem #PRU-31083

Problems Combinatorics

11–13 2.0

A class has more than 32, but less than 40 people. Every boy is friends with three girls, and every girl is friends with five boys. How many people are there in the class?

Problem #PRU-32068

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (area and volume)

11–13 3.0

A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?

Problem #PRU-32075

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (area and volume) Dissections, partitions, covers and tilings Various dissection problems

12–14 4.0

One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?

Problem #PRU-32135

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Combinatorics Systems of points and line segments Systems of points and line segments (other)

12–14 3.0

101 points are marked on a plane; not all of the points lie on the same straight line. A red pencil is used to draw a straight line passing through each possible pair of points. Prove that there will always be a marked point on the plane through which at least 11 red lines pass.

Problem #PRU-32992

Problems Geometry Solid geometry Regular polyhedra Main features and definitions of regular polyhedra Space transformations Projection Projection helps solve the problem Combinatorics Systems of points and line segments Systems of points and line segments (counterexamples)

13–13 2.0

a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?

b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?

Problem #PRU-34862

Problems Combinatorics

13–15 2.0

There are 30 ministers in a parliament. Each two of them are either friends or enemies, and each is friends with exactly six others. Every three ministers form a committee. Find the total number of committees in which all three members are friends or all three are enemies.