We solve problems

Problem #PRU-107702

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Features of dissection pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–16 3.0

The surface of a \(3\times 3\times 3\) Rubik’s Cube contains \(54\) squares. What is the maximum number of squares we can mark so that no marked squares share at least one vertex?

Make sure you show that both (a) you can achieve this maximum and (b) that you can’t do better than this maximum.

Problem #PRU-109748

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Polygons (other)

13–15 4.0

We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.

Problem #PRU-116087

Problems Geometry Plane geometry Triangles Similar triangles Circles Tangent lines and tangent circles Tangent lines to circles

15–16 4.0

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let \(C_{1}\) be the point of intersection, further from the vertex \(C\), of the circles constructed from the medians \(AM_{1}\) and \(BM_{2}\). Points \(A_{1}\) and \(B_{1}\) are defined similarly. Prove that the lines \(AA_{1}\), \(BB_{1}\) and \(CC_{1}\) intersect at the same point.

Problem #PRU-30290

Problems Number Theory Divisibility Odd and even numbers Geometry Plane geometry Polygons

10–12 2.0

a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.

b) What can be said about the case of a decagon?

Problem #PRU-30911

Problems Fun Problems Word Problems Solving problems by inequalities. Going through the cases. Geometry Solid geometry Volume Volume of a parallelepiped

11–12 2.0

Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?

Problem #PRU-32032

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality (other)

13–14 3.0

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.

Problem #PRU-32992

Problems Geometry Solid geometry Regular polyhedra Main features and definitions of regular polyhedra Space transformations Projection Projection helps solve the problem Discrete Mathematics 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-35238

Problems Geometry Plane geometry Triangles Types of triangles Equilateral triangle Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.

Problem #PRU-35765

Problems Discrete Mathematics Combinatorics Discrete geometry (other) Geometry Solid geometry Skeletons of multifaceted figures

13–14 2.0

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

Problem #PRU-56827

Problems Geometry Plane geometry Triangles

12–13 2.0

Prove that the bisectors of a triangle intersect at one point.