The numbers \(1, 2, \dots , 9\) are divided into three groups. Prove that the product of the numbers in one of the groups will always be no less than 72.
Some whole numbers are placed into a \(10\times 10\) table, so that the difference between any two neighbouring, horizontally or vertically adjacent, squares is no greater than 5. Prove that there will always be two identical numbers in the table.
Prove that in any group of 10 whole numbers there will be a few whose sum is divisible by 10.
You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.
Is it possible to place the numbers \(-1, 0, 1\) in a \(6\times 6\) square such that the sums of each row, column, and diagonal are unique?
A piece fell out of a book, the first page of which is the number 439, and the number of the last page is written with those same numbers in some other order. How many pages are in the fallen out piece?
Is it possible to fill a \(5 \times 5\) board with \(1 \times 2\) dominoes?
a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.
b) What can be said about the case of a decagon?
How many six-digit numbers exist, the numbers of which are either all odd or all even?
The Russian Chess Championship is made up of one round. How many games are played if 18 chess players participate?