The numbers \(\lfloor a\rfloor, \lfloor 2a\rfloor, \dots , \lfloor Na\rfloor\) are all different, and the numbers \(\lfloor 1/a\rfloor, \lfloor 2/a\rfloor,\dots , \lfloor M/a\rfloor\) are also all different. Find all such \(a\).
2022 points are selected from a cube, whose edge is equal to 13 units. Is it possible to place a cube with edge of 1 unit in this cube so that there is not one selected point inside it?
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 each created at least one maths problem, making 40 maths problems altogether. 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 \(26^{\circ}\).
Every integer from 1 to 64 is written in an \(8 \times 8\) table. Prove that no matter the order these numbers are written, there are two adjacent numbers which are at least \(5\) apart. (Numbers in cells which share a 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. Each person knew at least \(n\) 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. (\(n \geq 2\))