This is the boy who was asked by the teacher to stay back after school.
This is the boy whom the teacher asked to stay back after school.
This is the boy the teacher asked to stay back after school.
Are the sentences above correct ?
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Yes, the sentences are correct. In the first example pronoun "who" is in subjective case, in the second example it is in objective case. You can find some info and tips on their usage here: http://web.ku.edu/~edit/whom.html
As for the third example - pronouns who/whom can often be omitted. You will find more info on that on our related site. In short:
There is such a rule [concerning omission of who/whom], but it is not about whether the relative pronoun would be "who" or "whom": it is about whether or not the NP (noun phrase) which is the antecedent of the relative clause is the subject of that clause.