We solve problems

Problems Geometry Solid geometry Space transformations Isometry Point symmetry Reflectional symmetry

The opposite sides of a convex hexagon are pairwise equal and parallel. Prove that it has a centre of symmetry.

The symmetry axis of the polygon intersects its sides at points \(A\) and \(B\). Prove that the point \(A\) is either the vertex of the polygon or the middle of the side perpendicular to the axis of symmetry.

Problem #PRU-57914

Problems Geometry Plane geometry Plane transformations Isometry of the plane Rotations

12–14 2.0

Prove that a circle transforms into a circle when it is rotated.

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-58082

Problems Geometry Plane geometry Geometric inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Triangles Types of triangles Right-angled triangles Pythagorean theorem and its converse

12–14 4.0

A \(3\times 4\) rectangle contains 6 points. Prove that amongst them there will be two points, such that the distance between them is no greater than \(\sqrt5\).

Problem #PRU-58084

Problems Geometry Plane geometry Circles Circle, sector, segment, etc Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Pigeonhole principle (finite number of poits, lines etc.)

12–13 4.0

There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.

Problem #PRU-58089

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–15 4.0

What is the minimum number of points necessary to mark inside a convex \(n\)-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?

Problem #PRU-58313

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

Ten circles are marked on the circle. How many non-closed non-self-intersecting nine-point broken lines exist with vertices at these points?

Problem #PRU-60335

11–13 2.0

a) In Wonderland, there are three cities \(A\), \(B\) and \(C\). 6 roads lead from city \(A\) to city \(B\), and 4 roads lead from city \(B\) to city \(C\). How many ways can you travel from \(A\) to \(C\)?

b) In Wonderland, another city \(D\) was built as well as several new roads – two from \(A\) to \(D\) and two from \(D\) to \(C\). In how many ways can you now get from city \(A\) to city \(C\)?