Regular polygon = all angles have the same measure AND all sides have equal length.
So, is there a possibility to draw a regular pentagon with just a straightedge? (I think you may also call it a ruler.) If the answer is "yes", then how?
No compass is allowed!
Moreover, the construction must be absolutely precise.
It seems that it's quite a popular problem. I first read through it on a particular site; the suggested answers which some users provided were unfortunately incorrect (it's in Russian though, but I can provide the link to my problem if anyone wants). I tried googling, but I found no answer. Multiple searches, Wikipedia, Youtube—all in vain.
No. The construction of a regular pentagon entails the construction of a line segment of length $\sqrt{5}/2$. This is a big problem for your scenario in the following way: the base field of constructible numbers is $\mathbb{Q}$, which is closed under the addition/subtraction and multiplication/division; here, elements of $\mathbb{Q}$ are represented by line segments of rational length and the results of the aforementioned operations by their intersections. That is, if you're working purely with straightedge, then you cannot construct any non-rational lengths. But of course $\sqrt{5}/2$ is known to be irrational; thus you cannot construct it without using circles, which means that a regular pentagon cannot be constructed without a compass.