2. let $n$ be a natural number. Determine the number of subsets $A \subset\{1,2, \ldots, 2 n\}$ such that $x+y=2 n+1$ does not hold for any two elements $x, y \in A$.

## Solution:
The only solutions of the equation $x+y=2 n+1$ in $\{1,2, \ldots, 2 n\}$ are the pairs $(x, y)=(k, 2 n+1-k)$, where $1 \leq k \leq 2 n$. Now consider the disjoint decomposition

$$
\{1,2, \ldots, 2 n\}=\{1,2 n\} \cup\{2,2 n-1\} \cup \ldots \cup\{n-1, n+2\} \cup\{n, n+1\}
$$

The set $A$ fulfills the conditions if it contains at most one element from each of these $n$ two-element sets. There are exactly 3 ways to select at most one element from a two-element set, so according to the product rule there are $3^{n}$ such sets $A$.

## Remarks and scoring scheme: