26th Junior Balkan Mathematical Olympiad | JBMO 2022

Problem 4

We call an even positive integer $n$ nice if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$ . For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$ , $\{3, 6\}$ , $\{4, 5\}$ . Find the number of nice positive integers which are smaller than $3^{2022}$ .