B3. In the plane, 10 points are drawn, forming a regular triangular grid (see figure).

(a) How many equilateral triangles are there whose vertices lie on the points of this grid?

![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-09.jpg?height=214&width=257&top_left_y=156&top_left_x=1659)

(b) What is the minimum number of points we need to delete so that no equilateral triangle remains whose vertices are among the remaining points?

## Problems for 3rd 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 the 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 for set A.
![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-10.jpg?height=172&width=804&top_left_y=685&top_left_x=638)
B3. (a) There are exactly 15 triangles, of which 9 are small, 3 are medium-sized, 1 is large, and 2 are slanted. The images show one example of each of the four types of triangles.
![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-23.jpg?height=220&width=1058&top_left_y=630&top_left_x=176)

(b) It is certainly necessary to delete at least 3 points, otherwise at least 1 of the colored equilateral triangles in the images below would have all its vertices in non-deleted points.
![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-23.jpg?height=228&width=510&top_left_y=941&top_left_x=176)

If it were sufficient to delete 3 points, then the central point of the grid would not be deleted, and at least 1 of the deleted points would have to be a vertex of the large triangle. Thus, among the 6 points adjacent to the central point of the grid, at most 2 points would be deleted, and among the remaining 4 non-deleted points, there would be 2 that are adjacent. These 2 adjacent points would form a small equilateral triangle with the central point of the grid, with all vertices in non-deleted points. Therefore, it is necessary to delete at least 4 points. As shown in the image below, deleting 4 points is sufficient.

![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-23.jpg?height=223&width=255&top_left_y=1456&top_left_x=158)

Correctly counting 13 small, medium, and large equilateral triangles ... 1 point. Correctly counting 2 slanted equilateral triangles ... 1 point.

If the competitor only states the number 13 or 15 without explanation or calculation, they do not receive points.

Finding 3 triangles that do not share vertices and justification that we must remove at least one

![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-23.jpg?height=60&width=1731&top_left_y=2066&top_left_x=171)
Complete justification ............................................................................ 2 points. Correct construction with 4 removed points ............................................... 2 points.

If the competitor provides the correct construction but does not find all 15 triangles in part (a), they receive only 1 point for the construction.

## 66th Mathematical Competition for High School Students in Slovenia National Competition, April 23, 2022 Solutions to the problems for 3rd grade

![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-24.jpg?height=137&width=220&top_left_y=477&top_left_x=929)