We solve problems

Problems Methods Colouring Chessboard colouring Pigeonhole principle Pigeonhole principle (other)

A city in the shape of a triangle is divided into 16 triangular blocks, at the intersection of any two streets is a square (there are 15 squares in the city). A tourist began to walk around the city from a certain square and travelled along some route to some other square, whilst visiting every square exactly once. Prove that in the process of travelling the tourist at least 4 times turned by \(120^{\circ}\).

Problem #PRU-35771

Problems Methods Examples and counterexamples. Constructive proofs Calculus Functions of one variable. Continuity Periodicity and aperiodicity

15–16 4.0

Does a continuous function that takes every real value exactly 3 times exist?

Problem #PRU-57006

Problems Geometry Plane geometry Quadrilateral Circumscribed quadrilaterals

13–16 4.0

Prove that a convex quadrilateral \(ICEF\) can have a circle inscribed into it if and only if \(IC+EH = CE+IF\).

Problem #PRU-57808

Problems Geometry Plane geometry Plane transformations Parallel translation

13–15 4.0

Two circles of radius \(R\) touch at point \(B\). On one of them, point \(D\) is chosen and on the other point \(E\) is chosen. These points have a property of \(\angle DBE = 90^{\circ}\). Prove that \(DE = 2R\).

Problem #PRU-58082

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 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-58085

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Geometry Plane geometry Polygons Inscribed and circumscribed polygons Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 4.0

A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).

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 \(n\)-gon, so that at least one marked point always lies inside any triangle whose vertices are the vertices of the polygon?

Problem #PRU-58094

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Connection between the arc length, the angle, and the chord Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 4.0

Several chords are drawn through a unit circle. Prove that if each diameter intersects with no more than \(k\) chords, then the total length of all the chords is less than \(\pi k\).

Problem #PRU-58101

Problems Geometry Plane geometry Area Area of a circle, sector, segment Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

14–15 4.0

A square of side 15 contains 20 non-overlapping unit squares. Prove that it is possible to place a circle of radius 1 inside the large square, so that it does not overlap with any of the unit squares.