We solve problems

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 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 $\sqrt{5}$ .

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 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 $n$ -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-58093

Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–14 2.0

A plane contains $n$ straight lines, of which no two are parallel. Prove that some of the angles will be smaller than $180^{\circ} / n$ .

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 $π k$ .

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-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.

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-58220

Problems Dissections

13–14 2.0

Cut an arbitrary triangle into 3 parts and out of these pieces construct a rectangle.