We solve problems

Problems Geometry Plane geometry Polygons Regular polygons

Prove that the midpoints of the sides of a regular polygon form a regular polygon.

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations

14–16 3.0

Two circles \(c\) and \(d\) are tangent at point \(B\). Two straight lines intersecting the first circle at points \(D\) and \(E\) and the second circle at points \(G\) and \(F\) are drawn through the point \(B\). Prove that \(ED\) is parallel to \(FG\).

Problem #PRU-58080

Problems Combinatorics Geometry on grid paper Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–13 3.0

A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.

Problem #PRU-58081

Problems Geometry Plane geometry Triangles Types of triangles Equilateral triangle

12–14 3.0

There are 5 points inside an equilateral triangle with side of length 1. Prove that the distance between some two of them is less than 0.5.

Problem #PRU-58097

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Plane transformations Projection onto a line Orthogonal projection Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–16 3.0

Several circles, whose total length of circumferences is 10, are placed inside a square of side 1. Prove that there will always be some straight line that crosses at least four of the circles.

Problem #PRU-58107

Problems Geometry Plane geometry Area Finding the area by rearrangment Set theory and logic Set theory Inclusion-exclusion principle Methods Pigeonhole principle Pigeonhole principle (area and volume) Proof by contradiction

14–15 3.0

a) A square of area 6 contains three polygons, each of area 3. Prove that among them there are two polygons that have an overlap of area no less than 1.

b) A square of area 5 contains nine polygons of area 1. Prove that among them there are two polygons that have an overlap of area no less than \(\frac{1}{9}\).

Problem #PRU-60341

Problems Probability and statistics Probability theory Probability theory (other)

13–15 3.0

How many are there six-digit numbers that are divisible by \(5\)?

Problem #PRU-60364

Problems Algebra and arithmetic Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (other) Word problems Tables and tournaments Tournament tables

12–14 3.0

In a volleyball tournament teams play each other once. A win gives the team 1 point, a loss 0 points. It is known that at one point in the tournament all of the teams had different numbers of points. How many points did the team in second last place have at the end of the tournament, and what was the result of its match against the eventually winning team?

Problem #PRU-60365

Problems Combinatorics Geometry on grid paper Painting problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

An endless board is painted in three colours (each cell is painted in one of the colours). Prove that there are four cells of the same colour, located at the vertices of the rectangle with sides parallel to the side of one cell.

Problem #PRU-60367

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–15 3.0

On a plane, six points are given so that no three of them lie on the same line. Each pair of points is connected by a blue or red segment.

Prove that among these points three such points can be chosen so that all sides of the triangle formed by them will be of the same colour.