The table in the image is supposed to contain seven natural numbers, with each gray cell containing the product of the numbers in the two white cells adjacent to it. The numbers in the white cells are distinct, and the product of the numbers in the gray cells is 525.

What is the sum of the numbers in the gray cells? Find all possibilities.

(E. Patáková)

![](https://cdn.mathpix.com/cropped/2024_04_17_bf37c56bc0e241535d40g-1.jpg?height=89&width=1008&top_left_y=732&top_left_x=524)
We will mark the numbers in the white fields and use them to express the products in the gray fields — see the image.

| $a$ | $a \cdot b$ | $b$ | $b \cdot c$ | $c$ | $c \cdot d$ | $d$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |

The product of the numbers in the gray fields, in the established notation, is equal to $a \cdot b \cdot b \cdot c \cdot c \cdot d$, and this is supposed to be equal to 525. The only way to express this number as a product of six natural numbers is

$$
525=1 \cdot 1 \cdot 3 \cdot 5 \cdot 5 \cdot 7
$$

By comparing with the previous expression, we see that 1 and 5 must be in the inner white fields, while 3 and 7 must be in the outer ones. Two of the four possible ways to fill the table are shown in the following image. The remaining two fillings are axisymmetric to the ones given, thus giving the same sum of the numbers in the gray fields.

| 3 | 3 | 1 | 5 | 5 | 35 | 7 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 7 | 7 | 1 | 5 | 5 | 15 | 3 |

The possible sums of the numbers in the gray fields are $3+5+35=43$ and $7+5+15=27$.

Evaluation. 1 point for the factorization of the number 525 into 6 natural numbers; 2 points for finding one filling; 2 points for finding the second filling; 1 point for determining the correct sums.