It is known that a camera located at \(O\) cannot see the objects \(A\) and \(B\), where the angle \(AOB\) 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.
Two identical gears have 32 teeth. They were combined and 6 pairs of teeth were simultaneously removed. Prove that one gear can be rotated relative to the other so that in the gaps in one gear where teeth were removed the second gear will have whole teeth.
The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.
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.
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?
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 \(\lfloor \sqrt{\!\sqrt{x}}\rfloor = \lfloor \sqrt{\!\sqrt{x}}\rfloor\)?
In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.
A square \(ABCD\) contains 5 points. Prove that the distance between some pair of these points does not exceed \(\frac{1}{2} AC\).
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.
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?