A table of \(4\times4\) cells is given, in some cells of which a star is placed. Show that you can arrange seven stars so that when you remove any two rows and any two columns of this table, there will always be at least one star in the remaining cells. Prove that if there are fewer than seven stars, you can always remove two rows and two columns so that all the remaining cells are empty.
120 unit squares are placed inside a \(20 \times 25\) rectangle. Prove that it will always be possible to place a circle with diameter 1 inside the rectangle, without it overlapping with any of the unit squares.
All of the integers from 1 to 64 are written in an \(8 \times 8\) table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 5. (Numbers that are in cells which share a common side are called adjacent).
What is the largest amount of numbers that can be selected from the set 1, 2, ..., 1963 so that the sum of any two numbers is not divisible by their difference?
All of the integers from 1 to 81 are written in a \(9 \times 9\) table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 6. (Numbers that are in cells which share a common side are called adjacent.)
Prove that in a group of 11 arbitrary infinitely long decimal numbers, it is possible to choose two whose difference contains either, in decimal form, an infinite number of zeroes or an infinite number of nines.
A group of \(2n\) people were gathered together, of whom each person knew no less than \(n\) of the other people present. Prove that it is possible to select 4 people and seat them around a table so that each person sits next to people they know. (\(n \geq 2\))
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 \(2n\).