## Task A-1.5.

On the board, there are 2023 different real numbers. If each number on the board (simultaneously) is replaced by the sum of all the other numbers, the 2023 numbers on the board will be the same as at the beginning.

What values can the product of all the numbers on the board take at some point?
## Second Solution.

As in the previous solution, we conclude that if the numbers on the board after the first move are equal to the opposite values of the numbers written on the board at the beginning, and that the product does not change after each move.

Without loss of generality, let $x_{1}>x_{2}>\ldots>x_{2023}$. Then it holds that $-x_{2023}>$ $-x_{2022}>\ldots>-x_{1}$. Since the numbers written on the board at the beginning are equal to those after one move, it must be

$$
x_{1}=-x_{2023}, x_{2}=-x_{2022}, \ldots, x_{2023}=-x_{1}
$$

In particular, $x_{1012}=-x_{1012}$, or $x_{1012}=0$.

Since one of the numbers on the board is zero, we conclude that the product of these numbers is zero, and it will remain so after each move.

Note: The solution earns the first point from the first scoring scheme if it mentions in any form the equality that the sum of all numbers on the board before the swap is equal to the sum of all numbers on the board after the swap, or an analogous statement for the product.