## Task A-1.5.

Karlo and Lovro are playing the following game. Karlo will cut a paper of size $9 \times 9$ into rectangles of integer dimensions, where at least one dimension is 1. After that, Lovro will choose a natural number $k \in \{1, \ldots, 9\}$, and Karlo will give him as many coins as the total area of all rectangles of dimensions $1 \times k$ and $k \times 1$. Lovro will choose $k$ to maximize the number of coins he receives, while Karlo wants to save and give Lovro as few coins as possible.

Determine the smallest possible number of coins that Karlo will give to Lovro.
## Solution.

We claim that Karlo will give Lovro at least 12 coins.

1 point

Assume there is a way for Karlo to cut the paper such that Lovro has to give (strictly) fewer than 12 coins. By cutting the paper in this way, Karlo would create at most 11 rectangles of dimensions $1 \times 1$, five rectangles of dimensions $1 \times 2$, three rectangles of dimensions $1 \times 3$, two rectangles of dimensions $1 \times 4$, two rectangles of dimensions $1 \times 5$, and one rectangle each of dimensions $1 \times 6$, $1 \times 7$, $1 \times 8$, and $1 \times 9$.

The total area of all these rectangles would be

$$
11 \cdot 1 + 5 \cdot 2 + 3 \cdot 3 + 2 \cdot 4 + 2 \cdot 5 + 6 + 7 + 8 + 9 = 78
$$

Since $78 < 81$, Karlo would have to create at least one more rectangle to use the entire $9 \times 9$ paper. Therefore, it is impossible for Karlo to give Lovro fewer than 12 coins.

The following example shows that Karlo can cut the paper in such a way that he is sure Lovro will have to give at most 12 coins.

| 9 |  |  |  |  |
| :---: | :---: | :---: | :---: | :---: |
| 8 |  |  |  | 1 |
| 7 |  |  | ![](https://cdn.mathpix.com/cropped/2024_05_30_8e1d37b7107fd49224e9g-09.jpg?height=59&width=34&top_left_y=1252&top_left_x=1144) |  |
| 6 |  | 3 |  |  |
| 6 |  | 3 |  |  |
| 5 |  | 4 |  |  |
| 5 |  | 4 |  |  |
| 3 | 2 | 4 |  |  |
| 3 | 2 | 2 |  |  |

## COUNTY COMPETITION IN MATHEMATICS

2nd grade - high school - A variant
March 3, 2017.

## IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE TASK, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH IN AN APPROPRIATE MANNER.