## C1 BUL

A board $n \times n(n \geq 3)$ is divided into $n^{2}$ unit squares. Integers from 0 to $n$ included are written down: one integer in each unit square, in such a way that the sums of integers in each $2 \times 2$ square of the board are different. Find all $n$ for which such boards exist.


## Solution:

The number of the $2 \times 2$ squares in a board $n \times n$ is equal to $(n-1)^{2}$. All possible sums of the numbers in such squares are $0,1, \ldots, 4 n$. A necessary condition for the existence of a board with the required property is $4 n+1 \geq(n-1)^{2}$ and consequently $n(n-6) \leq 0$. Thus $n \leq 6$. The examples show the existence of boards $n \times n$ for all $3 \leq n \leq 6$.
![](https://cdn.mathpix.com/cropped/2024_06_05_07a1b610f48bd7c35addg-15.jpg?height=378&width=890&top_left_y=392&top_left_x=304)

| 6 | 6 | 6 | 6 | 5 | 5 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| 6 | 6 | 5 | 5 | 5 | 5 |
| 1 | 2 | 3 | 4 | 4 | 5 |
| 3 | 5 | 0 | 5 | 0 | 5 |
| 1 | 0 | 2 | 1 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 |

![](https://cdn.mathpix.com/cropped/2024_06_05_07a1b610f48bd7c35addg-15.jpg?height=98&width=228&top_left_y=810&top_left_x=271)

2015 points are given in a plane such that from any five points we can choose two points with distance less than 1 unit. Prove that 504 of the given points lie on a unit disc.

