We solve problems

Problems Geometry Solid geometry Lines and planes in space Methods Pigeonhole principle

An area of airspace contains clouds. It turns out that the area can be divided by 10 aeroplanes into regions such that each region contains no more than one cloud. What is the largest number of clouds an aircraft can fly through whilst holding a straight line course.

Problem #PRU-35575

Problems Methods Methods of calculus Continuity considerations Geometry Plane geometry Convex and non-convex figures Convex polygons Calculus Functions of one variable. Continuity Intermediate value theorem. Connectedness.

14–16 5.0

A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).

Problem #PRU-35598

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In a communication system consisting of 2001 subscribers, each subscriber is connected with exactly \(n\) others. Determine all the possible values of \(n\).

Problem #PRU-35632

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.

Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.

Problem #PRU-35643

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Plane geometry Polygons Regular polygons

13–14 4.0

2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.

Problem #PRU-35734

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

There are 20 students in a class, and each one is friends with at least 14 others. Can you prove that there are four students in this class who are all friends?

Problem #PRU-35758

Problems Geometry Plane geometry Triangles Incirlce and circumcircle of a triangle Polygons Pentagons Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Regular polygons

13–14 3.0

All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.

Problem #PRU-35765

Problems Combinatorics Discrete geometry Geometry Solid geometry Cross-sections, nets, and skeletons Skeletons of multifaceted figures

13–14 2.0

Prove that there is no polyhedron that has exactly seven edges.

Problem #PRU-37549

Problems Geometry Plane geometry Lines, segments, rays, and angles Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Triangles

11–13 3.0

Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.

Problem #PRU-52586

Problems Geometry Plane geometry Circles Inscribed angle An angle between two chords or two secants

13–14 2.0

A circle is divided up by the points \(A, B, C, D\) so that \({\smile}{AB}:{\smile}{BC}:{\smile}{CD}:{\smile}{DA} = 2: 3: 5: 6\). The chords \(AC\) and \(BD\) intersect at point \(M\). Find the angle \(AMB\).