The problem statement is:
Given $r \in (1,2)$, show that there exist infinitely many $n \in \mathbb{N}$ such that $\lceil{nr}\rceil$ is a power of 2.
I tried a few specific examples and I realized that in order for $\lceil nr \rceil = 2^k $,
$\frac{2^k - 1}{r} < n \leq \frac{2^k}{r}$ must occur for infinitely many natural numbers k, and specifically, both sides of the inequality must have different whole number parts, aka $\lfloor\frac{2^k - 1}{r}\rfloor < \lfloor\frac{2^k}{r}\rfloor$
I tried $r = 1.5$ and I spotted a pattern in the decimal expansion that allowed me to deduce that this inequality holds for all odd $k$, but I can't find something in general.