MathNet 2006 — Part 02 Question 99

Consider the set $A = \{1, 2, 3, 4, \ldots, 2^{n}\}$, $n \geq 2$. Find the number of the subsets $B$ of $A$, such that if the sum of two elements of $A$ is a power of $2$ then exactly one of them belongs to $B$.