$ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive

Question

$ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive. What is the smallest positive $ x$?

I tried ranges for each of cos(?)..that did not work

Answer 1

Assuming $x\in(0,1)$, let the binary representation of $x$ be $\overline{0.a_1a_2a_3\dots}_2$

The condition that $\cos(2\pi x)\le0$ is equivalent to that $(a_1,a_2)=(0,1)\text{ or }(1,0)$ (or technically $x=\frac34$, but note that $\frac34=0.10111\dots_2$)

Similarly the rest are equivalent to that $(a_2,a_3),(a_3,a_4),\dots,(a_5,a_6)\in\left\{(0,1),(1,0)\right\}$, and one may see that the smallest such $x$ is $0.010101_2$ or $\frac{21}{64}$.