24. $n$ people participate in a training competition consisting of 12 rounds. After each round, each participant receives a score $a_{k}\left(a_{k} \in \mathbf{N}_{+}, k=1,2, \cdots, n, a_{1}>a_{2}>\cdots>a_{n}\right)$ based on their rank $k$ in that round. After all the matches, the final ranking is determined by the sum of the scores each person has accumulated over the 12 rounds. Find the smallest positive integer $n$, for which it is possible to appropriately select $a_{1}, a_{2}, \cdots, a_{n}$ such that after the penultimate round, regardless of the rankings, at least 2 people have the possibility of becoming the ultimate champion.
24. Suppose one of the individuals (denoted as $A$) ranks first in the first 11 rounds, then his total score in the 12 rounds is no less than $11 a_{1} + a_{n}$. Assume that each of the other $n-1$ individuals ranks last in one of the first 11 rounds (as long as $n-1 \leqslant 11$, i.e., $n \leqslant 12$, this can be achieved), then the total score of the other $n-1$ individuals in the 12 rounds is no more than $a_{n} + 10 a_{2} + a_{1}$, because $11 a_{1} + a_{n} > a_{n} + 10 a_{2} + a_{1}$, which means $a_{1} > a_{2}$ always holds. Therefore, when $n \leqslant 12$, regardless of how $a_{1} > a_{2} > \cdots > a_{n}$ is chosen, it is possible that after 11 rounds, the champion is already uniquely determined. Hence, the smallest positive integer $n \geqslant 13$. On the other hand, when $n=13$, let $a_{1}=211, a_{2}=210, \cdots, a_{12}=200, a_{13}=1$. Suppose the individual with the highest total score after the first 11 rounds is $A$, and among the other 12 individuals, there is at least one (denoted as $B$) who did not rank last in any of the first 11 rounds. Thus, $A$'s score in the first 11 rounds does not exceed $11 a_{1} = 11 \times 211 = 2321$, while $B$'s score in the first 11 rounds is no less than $11 a_{12} = 11 \times 200 = 2200$. Therefore, $A$ can be at most 121 points more than $B$. $A$ clearly has the potential to win the championship. However, if $A$ ranks last in the 12th round, then $B$ will score at least 199 points more than $A$, thus surpassing $A$, meaning $A$ is not the champion, and therefore there is another individual who has the potential to become the champion after 11 rounds. In summary, the smallest value of $n$ is 13.