Problems

All integers from 1 to $2n$ are written in a row. Then, to each number, the number of its place in the row is added, that is, to the first number 1 is added, to the second – 2, and so on.

Prove that among the sums obtained there are at least two that give the same remainder when divided by $2n$.

What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?

Four lamps need to be hung over a square ice-rink so that they fully illuminate it. What is the minimum height needed at which to hang the lamps if each lamp illuminates a circle of radius equal to the height at which it hangs?

Is it possible to place the numbers $1,2,\dots 12$ around a circle so that the difference between any two adjacent numbers is 3, 4, or 5?

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?

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 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.

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.