On the New Year Eve, some of . . . . . . . . pupils stayed at . . . . . . . . school later than usual.

In the first blank, article "the" will be used because article 'the' is used to refer to something definite. In this sentence, speaker is referring to some specified number of pupils.
In the second blank, "no article or zero article" will be used because 'no article' is used with 'school, colleges' (if name of the school is not mentioned or there is no use of 'of').