We solve problems

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-79384

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 3.0

There are several squares on a rectangular sheet of chequered paper of size \(m \times n\) cells, the sides of which run along the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square contains another square within itself. What is the largest number of such squares?

Problem #PRU-79391

Problems Methods Mathematical induction Mathematical induction (other) Calculus Number sequences Number sequences (other)

14–15 4.0

At what value of \(k\) is the quantity \(A_k = (19^k + 66^k)/k!\) at its maximum? You are given a number \(x\) that is greater than 1. Is the following inequality necessarily fulfilled \(\lfloor \sqrt{\!\sqrt{x}}\rfloor = \lfloor \sqrt{\!\sqrt{x}}\rfloor\)?

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-79400

16–16 3.0

We consider a function \(y = f (x)\) defined on the whole set of real numbers and satisfying \(f (x + k) \times (1 - f (x)) = 1 + f (x)\) for some number \(k \ne 0\). Prove that \(f (x)\) is a periodic function.

Problem #PRU-79402

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Trigonometry Trigonometrical inequalities

15–16 4.0

Prove that the sequence \(x_n = \sin (n^2)\) does not tend to zero for \(n\) that tends to infinity.

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-79407

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–13 3.0

A square \(ABCD\) contains 5 points. Prove that the distance between some pair of these points does not exceed \(\frac{1}{2} AC\).

Problem #PRU-79426

Problems Geometry Plane geometry Triangles Types of triangles Equilateral triangle 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.

Problem #PRU-79445

Problems Combinatorics Painting problems Methods Proof by contradiction

12–14 3.0

The tracks in a zoo form an equilateral triangle, in which the middle lines are drawn. A monkey ran away from its cage. Two guards try to catch the monkey. Will they be able to catch the monkey if all three of them can run only along the tracks, and the speed of the monkey and the speed of the guards are equal and they can always see each other?