Official Explanation
The first and third blanks are highly related, so consider them first. “Apparent antagonism” exists between two people; are they really so hostile to one another? The two individuals also “derive inspiration and motivation from the other.” In other words, there is a “mutual benefit” to the seemingly hostile relationship, which otherwise might be “written off” as pure hatred, or “antipathy.” Neither “flagrancy” (the act of being flagrant, offensive) nor “hypocrisy” fits the meaning of hatred. For the third blank, both “worthless pride” and “tacit disagreement” are negative in nature; they do not convey a “benefit.” Now return to the second blank. The “apparent antagonism” can be “written off,” so “further scrutiny should render” clear or apparent the fact that there is actually a positive to this “rivalry.” The best fit is “patent” (obvious, evident). “Useless” and “spurious” (not genuine, insincere) do not fit the desired meaning.
Answer: B,E,H