We solve problems

Problems Geometry Solid geometry Visual geometry in space

How many times greater is the sum of the numbers in the hundred and first line of the Pascal triangle than the sum of the numbers in the hundredth line?

Problem #PRU-32903

Problems Geometry Solid geometry Visual geometry in space

12–12 2.0

Let’s put plus and minus signs in the 99th line of Pascal’s triangle. Between the first and second number there is a minus sign, between the second and the third there is a plus sign, between the third and the fourth there is a minus sign, then again a plus sign, and so on. Find the value of the resulting expression.

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-34875

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Geometry Plane geometry Polygons Hexagons Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Polygons Regular polygons

12–14 2.0

There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.

Problem #PRU-34905

Problems Geometry Plane geometry Area Area of a trapezium Area ratio Area ratio (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

13–14 4.0

9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.

Problem #PRU-34972

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

For which \(n > 3\), can a set of weights with masses of \(1, 2, 3, ..., n\) grams be divided into three groups of equal mass?

Problem #PRU-34980

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

14–15 3.0

A spherical planet is surrounded by 25 point asteroids. Prove, that at any given moment there will be a point on the surface of the planet from which an astronomer will not be able to see more than 11 asteroids.

Problem #PRU-35089

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

There are \(n\) cities in a country. Between each two cities an air service is established by one of two airlines. Prove that out of these two airlines at least one is such that from any city you can get to any other city whilst traveling on flights only of this airline.

Problem #PRU-35146

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In the secret service, there are \(n\) agents – 001, 002, ..., 007, ..., \(n\). The first agent monitors the one who monitors the second, the second monitors the one who monitors the third, etc., the nth monitors the one who monitors the first. Prove that \(n\) is an odd number.

Problem #PRU-35149

Problems Geometry Plane geometry Geometrical inequalities Area inequalities Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (area and volume)

13–15 3.0

In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.