We solve problems

Problem #PRU-116573

Problems Geometry Plane geometry Circles Diameter, chords, and secants Chords and secants (other) Polygons Regular polygons

15–17 3.0

What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular $n$ -gon?

Problem #PRU-110157

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations Circles Connection between the arc length, the angle, and the chord Calculus Derivative Derivative and multiple roots

14–16 5.0

The circles $σ_{1}$ and $σ_{2}$ intersect at points $A$ and $B$ . At the point $A$ to $σ_{1}$ and $σ_{2}$ , respectively, the tangents $l_{1}$ and $l_{2}$ are drawn. The points $T_{1}$ and $T_{2}$ are chosen respectively on the circles $σ_{1}$ and $σ_{2}$ so that the angular measures of the arcs $T_{1} A$ and $A T_{2}$ are equal (the arc value of the circle is considered in the clockwise direction). The tangent $t_{1}$ at the point $T_{1}$ to the circle $σ_{1}$ intersects $l_{2}$ at the point $M_{1}$ . Similarly, the tangent $t_{2}$ at the point $T_{2}$ to the circle $σ_{2}$ intersects $l_{1}$ at the point $M_{2}$ . Prove that the midpoints of the segments $M_{1} M_{2}$ are on the same line, independent of the positions of the points $T_{1}, T_{2}$ .

Problem #PRU-100516

Problems Geometry Plane geometry Area Area of a circle, sector, segment Circles

A maths teacher draws a number of circles on a piece of paper. When she shows this piece of paper to the young mathematician, he claims he can see only five circles. The maths teacher agrees. But when she shows the same piece of paper to another young mathematician, he says that there are exactly eight circles. The teacher confirms that this answer is also correct. How is that possible and how many circles did she originally draw on that piece of paper?

Problem #PRU-100596

Problems Geometry Plane geometry Area Area of a circle, sector, segment Circles

What is the ratio between the red and blue area? All shapes are semicircles.

Problem #PRU-100644

Problems Geometry Plane geometry Circles Triangles Types of triangles Right-angled triangles Pythagorean theorem and its converse

Find the area of the green annulus, if the marked pink segment has length $1$ .

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-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-78680

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Methods Mathematical induction Mathematical induction in geometry Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 3.0

On a circle of radius 1, the point $O$ is marked and from this point, to the right, a notch is marked using a compass of radius $l$ . From the obtained notch $O_{1}$ , a new notch is marked, in the same direction with the same radius and this is process is repeated 1968 times. After this, the circle is cut at all 1968 notches, and we get 1968 arcs. How many different lengths of arcs can this result in?

Problem #PRU-98408

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Diameter, chords, and secants Chords and secants (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Regular polygons

12–14 4.0

In a regular shape with 25 vertices, all the diagonals are drawn.

Prove that there are no nine diagonals passing through one interior point of the shape.