Problem 9.1. In a notebook, a triangular grid is drawn (see figure). Tanya placed integers at the nodes of the grid. We will call two numbers close if they are in adjacent nodes of the grid. It is known that

- the sum of all ten numbers is 43;
- the sum of any three numbers such that any two of them are close is 11.

What is the central number?

![](https://cdn.mathpix.com/cropped/2024_05_06_ce887724b4f1896529bag-1.jpg?height=458&width=519&top_left_y=671&top_left_x=467)
Answer: 10.

Solution. Let's denote the numbers by variables as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_ce887724b4f1896529bag-1.jpg?height=444&width=517&top_left_y=1270&top_left_x=468)

Then

\[
\begin{gathered}
a_{1}+a_{2}+a_{3}=b_{1}+b_{2}+b_{3}=c_{1}+c_{2}+c_{3}=11 \\
\left(a_{1}+a_{2}+a_{3}\right)+\left(b_{1}+b_{2}+b_{3}\right)+\left(c_{1}+c_{2}+c_{3}\right)+x=43
\end{gathered}
\]

From this, we get that \( x=10 \).

An example of a suitable arrangement of numbers is shown below.

![](https://cdn.mathpix.com/cropped/2024_05_06_ce887724b4f1896529bag-2.jpg?height=454&width=515&top_left_y=419&top_left_x=469)