$3-$ $[$ Algorithm Theory (miscellaneous). $\quad]$

Author: Berdikov A.

Sasha writes a sequence of natural numbers on the board. The first number $N>1$ is written in advance. New natural numbers he gets by subtracting from or adding to the last written number any of its divisors greater than 1. Can Sasha, for any natural $N>1$, eventually write the number 2011 on the board at some point?
The first method. By adding $N$ each time, we get 2011N. By subtracting 2011 each time, we get 2011.

The second method. If $N$ is odd, add $N$ and get an even number. By adding or subtracting twos from it, we get 4022. By subtracting 2011, we get 2011.

## Answer

In any case.

Author: $\underline{\text { Folklore }}$

In the cells of a $5 \times 5$ square table, the numbers 1 and -1 are arranged. It is known that the number of rows with a positive sum is greater than the number of rows with a negative sum.

What is the maximum number of columns in this table that can have a negative sum?

## Solution

Let's provide one possible example (see the table). The sum of the numbers in each of the top three rows is positive, while the sum of the numbers in each column is negative.

| 1 | 1 | 1 | -1 | -1 |
| :---: | :---: | :---: | :---: | :---: |
| 1 | -1 | 1 | -1 | 1 |
| -1 | 1 | -1 | 1 | 1 |
| -1 | -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 | -1 |

## Answer

5 columns.