B3. A table of size $1 \times 2022$ is given. The fields of the table are numbered sequentially from left to right with numbers from 1 to 2022. A black token is placed on field 1, and a white token on field 2. Two players, black and white, take turns playing the following game. The black player moves only the black token, and the white player only the white token. The player whose turn it is moves their token to an empty field on the table, in the direction towards the opponent's token. During this move, the player can either jump over the opponent's token or not. If the player whose turn it is cannot make a move, the game ends. The black player goes first. Determine the largest natural number $k$ such that the black player can force the white player to eventually move the white token to one of the fields numbered greater than or equal to $k$.

## Problems for 4th Grade

Time for solving: 180 minutes. Each problem in set A has exactly one correct answer. In set A, we will award two points for a correct answer and deduct one point for an incorrect answer. Write the answers for set A in the left table, leave the right table blank. The committee will consider only the answers written in the table when evaluating the answers for set A.
![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-14.jpg?height=174&width=808&top_left_y=684&top_left_x=634)
B3. We will show that $k=2021$. If the black player always moves their token to the field immediately to the right of the field with the white token, the white player must always move their token to the right and jump over the black token. Since the number of fields on the board is finite, sooner or later one of the players will not be able to make their move. If the black player cannot make a move, it means the white token is on field 2022. If the white player cannot make a move, it means the black token is on field 2022. However, since it is the white player's turn, the black player has just made their move, so the white token is on field 2021. In both cases, the white player is forced to place their token on a field with a number greater than or equal to 2021.

We also need to show that the white player can play in such a way that they never place their token on field 2022. The white player can choose the following strategy: If they have to move their token to the left, they move it to field 1 if it is empty, otherwise to field 2. If they have to move their token to the right, they move it to field 2021 if it is empty, otherwise to field 2020. We need to show that with this strategy, the white player is never forced to place their token on field 2022, meaning they can adhere to the strategy. It is possible that the black player might put the white player in a situation where they cannot make any allowed move, but this does not bother the white player as the game ends.

If the white player were forced to place their token on field 2022 in their turn, it would mean they have to move their token to the right and field 2022 is empty. If field 2021 were empty, the white player would move their token to field 2021 according to their strategy. Since they cannot do this, field 2021 must be occupied. This token cannot be white, as otherwise the white player would be moving their token to the left.

Therefore, the token on field 2021 is black. If field 2020 were empty, the white player would move their token to field 2020 according to their strategy. Since they cannot do this, field 2020 must be occupied by a white token. However, the black player has just made their move, which means that in the previous move of the white player, the black token was not on field 2021. Therefore, the white player, adhering to their strategy, would not have moved the white token to field 2020, as they would only do so if moving to the right and field 2021 were occupied by a black token, which we have just disproved. Such a situation could only arise if the white player did not adhere to their strategy.

Found strategy for the black player . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 points.

![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-27.jpg?height=51&width=1731&top_left_y=674&top_left_x=174)

Found strategy for the white player ............................................................... 2 points.

Proof that the white player can always adhere to the strategy and the recorded result $k=2021 \ldots \ldots .2$ points.

## 66th Mathematical Competition for High School Students in Slovenia

## National Competition, April 23, 2022  Solutions to Problems for 4th Year

![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-28.jpg?height=137&width=217&top_left_y=477&top_left_x=931)