2024 IMO Shortlist C1

Let $n$ be a positive integer. A class of $n$ students run $n$ races, in each of which they are ranked with no draws. A student is eligible for a rating $(a,b)$ for positive integers $a$ and $b$ if they come in the top $b$ places in at least $a$ of the races. Their final score is the maximum possible value of $a-b$ across all ratings for which they are eligible.

Find the maximum possible sum of all the scores of the $n$ students.

设 $n$ 为正整数。一个有 $n$ 名学生的班级参加 $n$ 场比赛，每场比赛都排出名次且没有并列。若学生在至少 $a$ 场比赛中进入前 $b$ 名，则称该学生有资格得到评分 $(a,b)$，其中 $a,b$ 为正整数。该学生的最终得分是在所有有资格得到的评分中 $a-b$ 的最大可能值。

求这 $n$ 名学生最终得分总和的最大可能值。