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?

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.

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?

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

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

The numbers $1,2,3,\dots ,99$ are written onto 99 blank cards in order. The cards are then shuffled and then spread in a row face down. The numbers $1,2,3,\dots ,99$ are once more written onto in the blank side of the cards in order. For each card the numbers written on it are then added together. The 99 resulting summations are then multiplied together. Prove that the result will be an even number.

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.