1) Everyone needs to pick up their books. (Arg!)
2) All the people in this room are required to submit your birthdays, and that will dictate which bus you ride on.
Sentences about groups of people tend to get complicated and become minefields for disagreement among subjects, verbs and other elements of predicates.
In sentence No. 1 above, a quick fix is to replace "their" with "his or her." Antecedents like "everyone, " "someone, " "everybody," "each," "either" are singular.
We used to treat "none" strictly as singular, because it is a contraction of "not one," but these days Harbrace 6a(7) says, "When used as subjects, "all," "any," "some," and "none" may take either a singular or a plural verb, generally depending on the context. Praise be.
In sentence No. 2, you might replace "that" with "those," to agree with birthdays.
On a different level of grammatical analysis, sentences like No. 2 can open up questions of logic: Is the sentence clear in saying that I am supposed to turn in my birthday, and not someone else's as well?
To achieve that clarity, you need the word "respective," or "respectively" in certain constructions.
For example, George Bush and Dick Cheney are the President and Vice President of the United States. Literally speaking, this says that they serve collectively in both offices. In this case, we might assert that the need for "respectively" is ridiculous---everybody knows that one is the President and other is the VP. But as skilled writers, editors, lawyers, (or SAT test takers), we must be aware that this is not what is actually expressed in the sentence.
One tip for sentences like No. 2--and ones that quickly get even more convoluted than this one--is to simplify the whole business by making the subject singular, if it's at all possible. In this case, that fix would be to turn "All the people in this room" to "Everyone in this room," and then "are required to submit their birthdays" to "is required to submit his or her birthday." This, in turn, makes the last part of the sentence more exact in what it's saying, in that it is actually the respective birthday of each individual that will determine which bus he or she rides on.
If you understand the paragraphs above, you may have a great future as a copy editor.
Note: Harbrace 6b(1) deals with agreements of antecedents and pronouns and has lots of great tips.