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Some students sit an examination with $50$ compulsory multiple choice questions, scoring $+2$ for a correct answer and $-1$ for an incorrect answer, with a minimum score of zero for the overall test.
Three of the students are discussing the possible marks:
Tyler says "Nobody will score the average mark".
Sadia says "Nobody will score higher than the average mark".
Joseph says "I will be the only person to score the average mark ".
Each student chooses their own definition of average from arithmetic mean, median and mode. Can you create a set of scores and choices of average for which they are all simultaneously correct in the two cases that there are an even number of examinees and an odd number of examinees?
Prove, in the two cases even/odd, that the 'choices' of averages made by Tyler, Sadia and Joseph are fixed, if it is possible that they are simultaneously correct.
Extension: Consider whether it is always possible simultaneously to meet these conditions for any number of students.