XLVI OM - III - Problem 1

Determine the number of subsets of the set $ \{1,2, \ldots , 2n\} $ in which the equation $ x + y = 2n+1 $ has no solutions.
A set $ A $, contained in the set $ \{1,2,\ldots,2n\} $, will be called good if it has the required property; that is, if the equation $ x + y = 2n + 1 $ has no solution $ x,y \in A $.
The set $ \{1,2,\ldots,2n\} $ is the union of the following disjoint two-element subsets (i.e., pairs of numbers):


the symbol $ P_j $ denotes the pair $ \{j, 2n+1-j\} $.
A subset $ A $ of the set $ \{1,2,\ldots,2n\} $ is good if and only if it does not contain any pair $ P_j $, or equivalently, if for every number $ j \in \{1,2,\ldots,n\} $ the intersection $ P_j \cap A $ is either the empty set, or the one-element set $ \{j\} $, or the one-element set $ \{2n+1-j\} $. A good set $ A $ is thus uniquely determined by choosing one of these three options for each $ j \in \{1,2,\ldots,n\} $. Hence, the number of good sets is $ 3^n $.