1. Natural numbers $a$, $b$, $c$ are placed in a circle as shown in the figure, with each number being a divisor of the sum of the two numbers adjacent to it. How many of the numbers $a$, $b$, $c$ can be different? (Josef Tkadlec)

![](https://cdn.mathpix.com/cropped/2024_04_17_c056f0bb4193640bf2e7g-2.jpg?height=266&width=300&top_left_y=244&top_left_x=1523)
SOLUTION. We will prove that necessarily $b=c$, thus among the numbers $a, b, c$ there can be at most two different ones. The count of 2 can be achieved, for example, in the case of numbers $a=2$ and $b=c=1$, which clearly have the required property.

Let us assume, therefore, that the natural numbers $a, b, c$ satisfy the conditions of the problem. This can be expressed by three conditions: