Can one bow before a king? Can one bow to a statue of a king?
For example, if one is visiting North Korea, may one bow to the statues as shown in this picture? Note that they are not fully prostrating themselves.
In North Korea the ruling class is (strangely) treated like a deity according to this Wikipedia article similar to Nevuchadzezzar and the idol he built. As such, bowing is assur (even without prostration - pishut yadaim veraglayim). Even removing one's hat in deference is assur (cf Yoreh Deah 150:3 Rema)
If one is bowing down to a person who is not treated as a diety and is not wearing religious items to which he is also bowing, there is no problem. Bowing to another person in the east (and frequently in the Torah) was the equivalent of today's handshake and mode of showing respect and is by itself permissible.
Related story from here
There is a chapter "Bowing" page 63 in the book "Letters to a Buddhist Jew" by Rabbis David Gottleib and Akiva Tatz that discusses at length this and similar issues.
One is allowed to bow before a king, even a secular king.
Indeed, the reason Mordechai did not bow before Haman (Esther 3:2) was only because he was wearing an idol (Esther Rabba 7:6) or had made himself into a god (Rashi ad loc). We see, therefore, that one is allowed to bow before a human leader, and this is not considered idolatry.
It is unclear if the leaders of North Korea consider themselves to be "divine." Officially, North Korea is atheist, and most of the population has no religion. However, some websites assert that they do consider themselves as gods.
If one is under duress, he is permitted to bow to a statue of a king, as long as the statue is not an idol. However, it is not clear if one may bow if he is not under duress.
(There is discussion in the Rishonim regarding this issue, based on Daniel chapter 3. See Maharsha in his Chiddushei Aggadot on Megilla 12a, Tosafot on Pesachim 53b, as well as most of the Rishonim on Ketuvot 33b and Sanhedrin 74a. See also this article.)