We solve problems

Problems Geometry Solid geometry Spheres Circle on a sphere Visual geometry in space

A globe has 17 parallels and 24 meridians. How many parts is the globe’s surface divided into? The meridian is an arc connecting the North Pole with the South Pole. A parallel is a circle parallel to the equator (the equator is also a parallel).

Problem #PRU-103888

Problems Methods Algebraic methods Proof by exhaustion Geometry Solid geometry Visual geometry in space

11–12 2.0

A square napkin was folded in half, the resulting rectangle was then folded in half again (see the figure). The resulting square was then cut with scissors (in a straight line). Could the napkin have been broken up a) into 2 parts? b) into 3 parts? c) into 4 parts? d) into 5 parts? If yes – illustrate such a cut, if not – write the word “no”.

Problem #PRU-103947

Problems Geometry Plane geometry Area

10–12 2.0

Find the area of the figures shown below.

Problem #PRU-104021

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Algebra and arithmetic Number theory. Divisibility Odd and even numbers Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?

b) The same question, but if 1 and \(-1\) are used instead.

Problem #PRU-104117

Problems Geometry Plane geometry Visual geometry

12–13 2.0

In a board, 20 pins are placed (see the picture). The distance between any adjacent pins is 1 inch. Pull a string of length 19 inches from the first pin to the second one, so that it goes through all the pins.

Problem #PRU-107702

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube 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 a vertex or are directly adjacent to another marked square?

Problem #PRU-108604

Problems Geometry Plane geometry Triangles Relationship of side lengths and angles of a triangle. Solving triangles. Ceva's theorem and Menelaus's theorem Calculus Integral Existence of a definite integral Vectors Law of polygon of vectors Similar triangles

13–14 3.0

On the sides \(AB\), \(BC\) and \(AC\) of the triangle \(ABC\) points \(P\), \(M\) and \(K\) are chosen so that the segments \(AM\), \(BK\) and \(CP\) intersect at one point and \[\vec{AM} + \vec{BK}+\vec{CP} = 0\] Prove that \(P\), \(M\) and \(K\) are the midpoints of the sides of the triangle \(ABC\).

Problem #PRU-109748

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons 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-110207

Problems Methods Pigeonhole principle Pigeonhole principle (other) Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality algebra problems

12–14 3.0

Natural numbers from 1 to 200 are divided into 50 sets. Prove that in one of the sets there are three numbers that are the lengths of the sides of a triangle.

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.