I believe you are saying that I interpreted the problem correctly.
To make things simple, we can suppose that the correct answer for each question is T.
I mentioned the pigeonhole principle, which you can look up. It's often used in fancy ways, but often I just think this way: If your goal is to find such a pair every time, my goal as your nemesis is to prevent you from doing so, by arranging the answers so it can't happen. That is, I want to find a counterexample to your claim (a set of answers to the 5 questions by N people, for which no pair got a T for each question). I'm hoping to find something that will prevent me from doing so.
It's easy to show that I can't find a counterexample for 3 people. If each row is a question and each column is a person, each row has to have at least two T's in it, because at least half must be correct. I'll try to make sure that every pair of columns has FF in some row. But I
can't put FF in
any row! There can only be one F. So it has to look something like this at worst:
FTT
TFT
TTF
FTT
TFT
Here
every pair fits your requirement; for example, the first and last columns taken alone are
FT
TT
TF
FT
TT
and at least one of them got T in every row.
Try doing this with 4 people, and see what happens. If you can find a counterexample, you've proved your claim wrong; if you can't, try to see why you couldn't, and that may be the proof you're asking for.