Japan, china, and korea were friendly nations who shared culture and customs. rival nations who came into frequent conflict. neighboring countries who shared a common history. countries with very dissimilar culture and customs.

Well, I'm not too sure there is one good answer to this question. 
two options are clearly false: friendly nations who shared culture and customs. and countries with very dissimilar culture and customs.- they do share culture but had way too many conflicts to be called friendly
 
The other two options are both partially true: 
rival nations who came into frequent conflict. - yes, they did come into frequent conflict, but Korea was rarely a rival there, it was rather subdued. But I still think that this is the best answer/ 

neighboring countries who shared a common history. - this could also be argued, depending on what you accept to be neighbouring countries: they Japan is an island and it can share only borders over water, not land with the others- this is also a good answer.