Question 40
Alberto chooses 2022 integers $a_{1}, a_{2}, \ldots, a_{2022}$ (not necessarily positive and not necessarily distinct) and arranges them in a $2022 \times 2022$ table so that the cell $(i, j)$ contains the number $a_{k}$, with $k$ equal to the maximum of $i$ and $j$, as in the following figure (in which, for readability, we have written $a_{n}$ for $a_{2022}$).
| $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | | $a_{n}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $a_{2}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | | $a_{n}$ |
| $a_{3}$ | $a_{3}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | | $a_{n}$ |
| $a_{4}$ | $a_{4}$ | $a_{4}$ | $a_{4}$ | $a_{5}$ | | $a_{n}$ |
| $a_{5}$ | $a_{5}$ | $a_{5}$ | $a_{5}$ | $a_{5}$ | | $a_{n}$ |
| . . . | | | | | | $a_{n}$ |
| $a_{n}$ | $a_{n}$ | $a_{n}$ | $a_{n}$ | $a_{n}$ | $a_{n}$ | $a_{n}$ |
Barbara does not know the numbers chosen by Alberto, but she knows how they were arranged in the table. Having fixed an integer $k$ with $1 \leq k \leq 2022$, Barbara wants to determine the value of $a_{k}$, while she is not interested in determining the values of the other $a_{i}$ with $i \neq k$. To do so, Barbara may ask Alberto one or more questions, in each of which she asks Alberto for the sum of the numbers contained in the cells of a "path", where by "path" we mean an ordered list of cells with the following properties:
- the path starts at the top-left cell and ends at the bottom-right cell,
- the cells of the path are all distinct,
- two consecutive cells of the path always share a side.
Determine, as $k$ varies, the minimum number of questions Barbara needs in order to determine $a_{k}$.
| $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | | $a_{n}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $a_{2}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | | $a_{n}$ |
| $a_{3}$ | $a_{3}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | | $a_{n}$ |
| $a_{4}$ | $a_{4}$ | $a_{4}$ | $a_{4}$ | $a_{5}$ | | $a_{n}$ |
| $a_{5}$ | $a_{5}$ | $a_{5}$ | $a_{5}$ | $a_{5}$ | | $a_{n}$ |
| . . . | | | | | | $a_{n}$ |
| $a_{n}$ | $a_{n}$ | $a_{n}$ | $a_{n}$ | $a_{n}$ | $a_{n}$ | $a_{n}$ |
Barbara does not know the numbers chosen by Alberto, but she knows how they were arranged in the table. Having fixed an integer $k$ with $1 \leq k \leq 2022$, Barbara wants to determine the value of $a_{k}$, while she is not interested in determining the values of the other $a_{i}$ with $i \neq k$. To do so, Barbara may ask Alberto one or more questions, in each of which she asks Alberto for the sum of the numbers contained in the cells of a "path", where by "path" we mean an ordered list of cells with the following properties:
- the path starts at the top-left cell and ends at the bottom-right cell,
- the cells of the path are all distinct,
- two consecutive cells of the path always share a side.
Determine, as $k$ varies, the minimum number of questions Barbara needs in order to determine $a_{k}$.