Minimum questions is:
Student 1 answers 1, 2, 3, and 4 correctly
Student 2 answers 3, 4, 5, and 6 correctly
Student 3 answers 1, 3, 5, and 7 correctly
✔ Number of questions is odd.
✔ Each question is answered correctly at least once.
✔ Every student answers four correctly; 4 is an even number.
✔ Any two students answer an even number of questions -
1+2 answer 1, 2, 3, 4, 5, and 6 - six questions is even.
1+3 answer 1, 2, 3, 4, 5, and 7 - six questions is even.
2+3 answer 1, 3, 4, 5, 6, and 7 - six questions is even.
It can't be less, because -
The number of questions must be odd.
• It can't be 1, as then the only even number of questions students could answer would be zero, and if all answer zero, not all questions get answered.
• It can't be 3, as then there are only three combinations of an even number of correct answers available to any student (we exclude "0"): 1+2, 1+3, and 2+3. At least two of these are needed to cover all 3 questions, and there is no way to pick two without there being a pair of students whose combined answers is 3, and thus not even.
• It can't be 5; each student could have two or four answers, but any combination of answerers that includes a student with four correct answers means that student plus a student with the fifth answer would make those two students a pair whose combined answers is 5, and thus not even - so each student can answer at most 2. You then run into a parallel scenario as for 3 questions, where it is always possible to find at least one pair of two answers out of any set of pairs that covers all of 1 to 5 that overlaps on one value, resulting in a pair of students whose combined answers is 3, and thus not even.