1. An exam consists of $m$ questions, with $n$ students participating, where $m, n \geqslant 2$ are given integers. The scoring rule for each question is: if exactly $x$ students fail to answer the question correctly, then each student who answers the question correctly gets $x$ points, and those who fail to answer correctly get zero points. Each student's total score is the sum of their scores on the $m$ questions. The total scores of all students, arranged in descending order, are $p_{1} \geqslant p_{2} \geqslant \cdots \geqslant p_{n}$. Find the maximum possible value of $p_{1}+p_{n}$.
(2013, National High School Mathematics Joint Competition)
Hint: First consider the extreme case: one student answers all questions correctly, and the other $n-1$ students answer all questions incorrectly.
Then $p_{1}+p_{n}=p_{1}=\sum_{k=1}^{m}(n-1)=m(n-1)$.
Next, prove: $p_{1}+p_{n} \leqslant m(n-1)$.