Note that if you turn over a sheet on which numbers are written, then the digits 0, 1, 8 will not change and the digits 6 and 9 will switch places, whilst the others will lose their meaning. How many nine-digit numbers exist that do not change when a sheet is turned over?
30 pupils in years 7 to 11 took part in the creation of 40 maths problems. Every possible pair of pupils in the same year created the same number of problems. Every possible pair of pupils in different years created a different number of problems. How many pupils created exactly one problem?
The number \(A\) is divisible by \(1, 2, 3, \dots , 9\). Prove that if \(2A\) is presented in the form of a sum of some natural numbers smaller than 10, \(2A= a_1 +a_2 +\dots +a_k\), then we can always choose some of the numbers \(a_1, a_2, \dots , a_k\) so that the sum of the chosen numbers is equal to \(A\).
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.
You are given \(7\) straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than \(26^{\circ}\).
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?
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.