B3. We have a sheet of graph paper of size $7 \times 7$ squares and an equally large plastic graph stencil, on which some squares are green, and the rest are transparent. If we place the stencil on the sheet of paper so that the sides of the stencil align with the sides of the paper, the grid pattern on the stencil matches the grid pattern on the paper. If we choose any square on the paper and color it red, then we can place the stencil on the paper so that its sides align with the sides of the paper and the red square on the paper is covered by one of the green squares on the stencil (the stencil can be rotated and flipped over as needed). What is the minimum number of green squares on the stencil?

67th Mathematical Competition for High School Students in Slovenia National Competition, April 22, 2023

# Problems for 3rd Year 

Time for solving: 180 minutes. Each problem in section A has exactly one correct answer. In section A, we will award two points for the correct answer, and deduct one point for an incorrect answer. Write the answers for section A in the left column of the answer sheet, and leave the right column blank. The committee will consider only what is written on the answer sheets, marked with the contestant's code.
B3. A template can be placed on paper in 8 different ways; in 4 ways with the front side facing up, and in 4 ways with the back side facing up. We transition between different placements by rotating the template by $90^{\circ}$ around its center and by flipping the template over one of its 4 axes of symmetry (which corresponds to reflecting over that axis). Therefore, the same green square on the template can cover those squares on the paper that we reach with these operations. Let's label the squares on the paper with numbers, as shown in the figure.

| 1 | 2 | 3 | 4 | 3 | 2 | 1 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 2 | 5 | 6 | 7 | 6 | 5 | 2 |
| 3 | 6 | 8 | 9 | 8 | 6 | 3 |
| 4 | 7 | 9 | 10 | 9 | 7 | 4 |
| 3 | 6 | 8 | 9 | 8 | 6 | 3 |
| 2 | 5 | 6 | 7 | 6 | 5 | 2 |
| 1 | 2 | 3 | 4 | 3 | 2 | 1 |

All sets of squares with the same number are symmetric with respect to all 4 axes of symmetry of the paper and remain unchanged if rotated by $90^{\circ}$ around the center of the paper. Therefore, one green square on the template can only cover one set of squares with the same number. Thus, the template must have at least 10 green squares. This is sufficient because each set of squares with the same number can be covered by one green square on the template (using the operations mentioned above). Therefore, if the template has 10 green squares, each covering a square with a different number, as shown in the example, then we can cover any square on the paper with such a template.

- Determining that a square has 8 symmetry operations (can also be implicit) [1 point]
- Determining that one green square can cover all squares that it maps to with the above operations [2 points]
- Determining that the set of all 49 squares can be divided into subsets, where in each subset the squares can be mapped to each other with some symmetry operation [2 points]
- Determining that we must select at least one square from each of these subsets [1 point]
- Determining that there are 10 such squares [1 point]

67th Mathematical Competition for High School Students in Slovenia National Competition, April 22, 2023

# Solutions to problems for 3rd grade

![](https://cdn.mathpix.com/cropped/2024_06_07_95af3fbbcea2fb458cf3g-12.jpg?height=137&width=232&top_left_y=451&top_left_x=929)