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.

The White Rook pursues a black horse on a board of $3\times 1969$ cells (they walk in turn according to the usual rules). How should the rook play in order to take the horse? White makes the first move.

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 a set there are 100 weights, each two of which differ in mass by no more than 20 g. Prove that these weights can be put on two cups of weighing scales, 50 pieces on each one, so that one cup of weights is lighter than the other by no more than 20 g.

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.

Peter bought an automatic machine at the store, which for 5 pence multiplies any number entered into it by 3, and for 2 pence adds 4 to any number. Peter wants, starting with a unit that can be entered free of charge to get the number 1981 on the machine number whilst spending the smallest amount of money. How much will the calculations cost him? What happens if he wants to get the number 1982?