Problems

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.

Airlines connect pairs of cities. How can you connect 50 cities with the fewest number of airlines so that from every city you can get to any other city by taking at most two flights?

In a corridor of length 100 m, 20 sections of red carpet are laid out. The combined length of the sections is 1000 m. What is the largest number there can be of distinct stretches of the corridor that are not covered by carpet, given that the sections of carpet are all the same width as the corridor?

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 $\lfloor h\times {2021}^{n-1}\rfloor$?

A monkey escaped from it’s cage in the zoo. Two guards are trying to catch it. The monkey and the guards run along the zoo lanes. There are six straight lanes in the zoo: three long ones form an equilateral triangle and three short ones connect the middles of the triangle sides. Every moment of the time the monkey and the guards can see each other. Will the guards be able to catch the monkey, if it runs three times faster than the guards? (In the beginning of the chase the guards are in one of the triangle vertices and the monkey is in another one.)

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.

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.