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?

Two lines on the plane intersect at an angle $\alpha$. 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 $\alpha$ the value $\alpha /\pi$ is a rational number.

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

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 $\lfloor \sqrt{\sqrt{x}}\rfloor =\lfloor \sqrt{\sqrt{x}}\rfloor$?

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.

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

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?

The judges of an Olympiad decided to denote each participant with a natural number in such a way that it would be possible to unambiguously reconstruct the number of points received by each participant in each task, and that from each two participants the one with the greater number would be the participant which received a higher score. Help the judges solve this problem!

Find the minimum for all $\alpha$, $\beta$ of the maximum of the function $y(x)=|\cos x+\alpha \cos 2x+\beta \cos 3x|$.