DE-BUNDESWETTBEWERB 2024 Problem 6

Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties: (1) No number occurs more than once in the sequence. (2) The sum of two different elements of the sequence is never a power of two. (3) For all positive integers $n$ , we have $a_n<r \cdot n$ .