The number \(n\) has the property that when it is divided by \(q^2\) the remainder is smaller than \(q^2 / 2\), whatever the value of \(q\). List all numbers that have this property.
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?
Does there exist a number \(h\) such that for any natural number \(n\) the number \(\lfloor h \times 2021^n\rfloor\) is not divisible by \(\lfloor h \times 2021^{n-1}\rfloor\)?
In the country of Mara there are several castles. Three roads lead from each castle. A knight left from one of the castles. Traveling along the roads, he turns from each castle standing in his way, either to the right or to the left depending on the road on which he came. The knight never turns to the side which he turned before it. Prove that one day he will return to the original castle.
Prove that for every convex polyhedron there are two faces with the same number of sides.
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.
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.