A \(3\times 3\) square is filled with the numbers \(-1, 0, +1\). Prove that two of the 8 sums in all directions – each row, column, and diagonal – will be equal.
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.
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?
The Russian Chess Championship is made up of one round. How many games are played if 18 chess players participate?
Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?
If a salary is first increased by 20%, and then reduced by 20%, will the salary paid increase or decrease as a result?
If a class of 30 children is seated in the auditorium of a cinema there will always be at least one row containing no fewer than two classmates. If we do the same with a class of 26 children then at least three rows will be empty. How many rows are there in the cinema?
Two grandmasters in turn put rooks on a chessboard (one turn – one rook) so that they cannot capture each other. The person who cannot put a rook on the chessboard loses. Who will win with the game – the first or second grandmaster?
In a tournament by the Olympic system (the loser is eliminated), 50 boxers participate. What is the minimum number of matches needed to be conducted in order to identify the winner?
In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).