Question 10
Suppose there are $m$ questions in an examination attended by $n$ students, where $m, n \ge 2$ are given natural numbers. The marking rule for each question is as follows: if there are exactly $x$ students failing to answer the question correctly, then they will each get 0 marks, and those who answer it correctly will each get $x$ marks. The total marks of a student are the sum of marks he/she gets from the $m$ questions. Now rank the total marks of the $n$ students as $p_1 \ge p_2 \ge \cdots \ge p_n$. Find the maximum possible value of $p_1 + p_n$.