\section*{Task 1 - 141031}

In the schema, the letters should be replaced by digits so that a correctly solved addition problem is formed. The same letters should be replaced by the same digits, and different letters should be replaced by different digits.

Provide all solutions for this! (A and Ä are considered different letters.)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1347.jpg?height=135&width=363&top_left_y=458&top_left_x=1395)
}

By using the letters appearing above as variable names with their corresponding numerical values, we can immediately set up the equation, since \(\ddot{A}=1\) must hold:

\[
2(1000 A+100 R+10 Z+T)=10000+1000 R+100 Z+10 T+E
\]

which, after a simple rearrangement, yields

\[
100 R+10 Z+T=250(A-5)-\frac{E}{8}
\]

From this, we can immediately conclude that \(E\) must be divisible by 8, and since \(E=0\) would immediately lead to the contradiction \(T=E\), the only remaining possibility is \(E=8\).

Since \(100 R+10 Z+T\) is certainly positive, the only remaining possibilities for \(A\) are \(A \in\{6,7,8,9\}\), where the cases \(A=7\) and \(A=9\) can be immediately ruled out as they would lead to the contradiction \(Z=T=9\), and \(A=E=8\) is of course not possible. This leaves only the case

\[
A=6,100 R+10 Z+T=249
\]

which is indeed a solution to the problem:

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1347.jpg?height=124&width=310&top_left_y=1454&top_left_x=879)