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Problem #PRU-78270

Problems Geometry Plane geometry Area Area of curvilinear shapes Methods Pigeonhole principle Pigeonhole principle (area and volume)

13–15 4.0

120 unit squares are placed inside a $20 \times 25$ rectangle. Prove that it will always be possible to place a circle with diameter 1 inside the rectangle, without it overlapping with any of the unit squares.

Problem #PRU-78475

Problems Geometry Plane geometry Lines, segments, rays, and angles

13–14 3.0

You are given $7$ straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than $26^{\circ}$ .

Problem #PRU-78675

Problems Geometry Plane geometry Vectors Visual geometry

11–13 2.0

Two lines on the plane intersect at an angle $α$ . On one of them there is a flea. Every second it jumps from one line to the other (the point of intersection is considered to belong to both straight lines). It is known that the length of each of her jumps is 1 and that she never returns to the place where she was a second ago. After some time, the flea returned to its original point. Prove that for the angle $α$ the value $α / π$ is a rational number.

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-78768

Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Geometry Plane geometry Polygons Sum of internal and external angles of polygons

12–14 4.0

It is known that a camera located at $O$ cannot see the objects $A$ and $B$ , where the angle $A O B$ is greater than $179^{\circ}$ . 1000 such cameras are placed in a Cartesian plane. All of the cameras simultaneously take a picture. Prove that there will be a picture taken in which no more than 998 cameras are visible.

Problem #PRU-79248

Problems Geometry Solid geometry Convex solids Graph theory Polyhedra and polygons in space

14–16 3.0

Prove that for every convex polyhedron there are two faces with the same number of sides.

Problem #PRU-79322

Problems Geometry Solid geometry Spheres Circle on a sphere Methods Pigeonhole principle Pigeonhole principle (area and volume) Extremal principle Smallest or largest distance (length)

14–16 4.0

A spherical sun is observed to have a finite number of circular sunspots, each of which covers less than half of the sun’s surface. These sunspots are said to be enclosed, that is no two sunspots can touch, and they do not overlap with one another. Prove that the sun will have two diametrically opposite points that are not covered by sunspots.

Problem #PRU-79399

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Plane geometry Polygons Regular polygons Quadrilateral Trapeziums

13–15 4.0

In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.

Problem #PRU-79403

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality (other) Visual geometry

13–15 4.0

In a square with side length 1 there is a broken line, which does not self-intersect, whose length is no less than 200. Prove that there is a straight line parallel to one of the sides of the square that intersects the broken line at a point no less than 101 units along the line.

Problem #PRU-79426

Problems Geometry Plane geometry Triangles Types of triangles Equilateral triangle Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

A white plane is arbitrarily sprinkled with black ink. Prove that for any positive $l$ there exists a line segment of length $l$ with both ends of the same colour.