Problems

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.

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.

The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.

A conference was attended by a finite group of scientists, some of whom are friends. It turned out that every two scientists, who have an equal number of friends at the conference, do not have friends in common. Prove that there is a scientist who has exactly one friend among the conference attendees.

There are several squares on a rectangular sheet of chequered paper of size $m\times n$ cells, the sides of which run along the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square contains another square within itself. What is the largest number of such squares?