The pair’s apparent antagonism could easily be written off as (i) Antipathy pure and simple. What does that mean ?

The pair’s apparent antagonism or rivalry between two, for instance, tennis players such as the historic rivalry between McEnroe and Borg.

Indeed, it was friendship and mutual respect. Apparently, it appeared like antipathy towards each other

Regards
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Can someone explain Blank 2 ?

patent = licence = certificate the fact ...............

The other two are out of scope.

Ask for further assistance if you need.

Regards
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The pair’s apparent antagonism could easily be written off as (i) ________ pure and simple, but further scrutiny should render (ii) ________ the fact that the rivalry also confers a fair amount of (iii) _________, insofar as it provides each an opportunity to derive inspiration and motivation from the other.

It is better to attack this Text Completion Triple Blank Question backwards.

Start from the third blank.

If the rivalry provides each an opportunity to derive inspiration and motivation from each other, then it is conferring mutual benefit on them.

Now let us go to the first and second blanks.

The sentence talks about the pair's apparent antagonism, hinting that there is no real antagonism. Also the phrase "could easily be written off" suggests that is the lazy and incorrect to suggest an antagonism and there is more to this than meets the eye. And then it goes on to say that further scrutiny reveals the fact that rivalry confers a fair amount of mutual benefit.

"further scrutiny should render (ii) ________ the fact that the rivalry also confers a fair amount of mutual benefit". My own word for the blank is "transparent".

The word from the choice list is patent. The word spurious would distort the intended meaning and useless is unsupported by evidence in the sentence.

Now let us go to the first blank.

Since deeper reflection reveals that the rivalry confers mutual benefit as each derive inspiration and motivation from each other, the simpler view would be that there is antipathy between the two. The word antipathy conveying the exact opposite sense of mutual benefit.

There is nothing to support dissimulation which is the concealment of one's thoughts, feelings, or character; pretence.

And flagrancy or atrociousness would indicate real and not apparent antagonism - and hence needs to be rejected.
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misread the second blank