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Several football teams are taking part in a football tournament, where each team plays every other team exactly once. Prove that at any point in the tournament there will be two teams who have played exactly the same number of matches up to that point.

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.

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.

There are bacteria in a glass. After a second each bacterium divides in half to create two new bacteria. Then after another second these bacteria divide in half, and so on. After a minute the glass is full. After how much time will the glass be half full?