Problems

30 teams are taking part in a football championship. Prove that at any moment in the contest there will be two teams who have played the same number of matches up to that moment, assuming every team plays every other team exactly once by the end of the tournament.

Several pieces of carpet are laid along a corridor. Pieces cover the entire corridor from end to end without omissions and even overlap one another, so that over some parts of the floor lie several layers of carpet. Prove that you can remove a few pieces, perhaps by taking them out from under others and leaving the rest exactly in the same places they used to be, so that the corridor will still be completely covered and the total length of the pieces left will be less than twice the length corridor.

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 $2n$.

In draughts, the king attacks by jumping over another draughts-piece. What is the maximum number of draughts kings we can place on the black squares of a standard $8\times 8$ draughts board, so that each king is attacking at least one other?

In order to encrypt telegraph signals it is necessary to divide every possible 10 character ‘word’ – an arrangement of 10 dots and dashes – into two groups, so that any two words in the same group differed by no fewer than three characters. Find a method of doing this or prove that no such method exists.

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?