Zuzka wrote the integers from 1 to 9, each exactly once, into nine fields on the following image. The ratio of the sums of the numbers written in the circles, triangles, and hexagons was $2: 3: 6$.

Determine which number could have been written in the top triangle; find all possibilities.

(E. Novotná)

![](https://cdn.mathpix.com/cropped/2024_04_17_7f4ac78ad31cb883e531g-2.jpg?height=391&width=412&top_left_y=547&top_left_x=822)
The sum of all numbers entered in the fields on the image is equal to

$$
1+2+\ldots+9=45
$$

The numbers in the triangles and the numbers in the hexagons are mutually different, and the ratio of their sums is $3: 6=1: 2$. Therefore, the sum of the numbers in the triangles is 15 and the sum of the numbers in the hexagons is 30. Since the ratio of the sums of the numbers in the circles and in the triangles is $2: 3$, the sum of the numbers in the circles is 10.

There are four circles on the image, so the numbers in the circles must be 1, 2, 3, and 4 (any other quartet would have a sum greater than 10). Of these four numbers, two are also contained in the triangles - in the remaining top triangle, there must be such a number that the sum of these three numbers is 15. The largest possible sum of the numbers in the two bottom triangles is 7, so the smallest possible number in the top triangle is 8. At the same time, the number in the top triangle cannot be greater than 9:

- if the number in the top triangle is 8, then the numbers in the two bottom triangles must be 3 and 4,
- if the number in the top triangle is 9, then the numbers in the two bottom triangles must be 2 and 4.

In both cases, the other numbers can be easily filled in so that all the requirements in the problem are met, see the images. The number 8 or 9 could have been written in the top triangle.
![](https://cdn.mathpix.com/cropped/2024_04_17_7f4ac78ad31cb883e531g-2.jpg?height=380&width=992&top_left_y=2049&top_left_x=540)

Scoring suggestion. 2 points for expressing the sums of the numbers in the triangles and hexagons; 1 point for expressing the quartet of numbers in the circles; 2 points for determining the two possible numbers in the top triangle; 1 point for completing the image and verification.