14.5.41 $\star \star$ In a competition lasting $k$ days, $n(\geqslant 2)$ players participate. Each day, the possible scores for the players are $1,2,3, \cdots, n$, and no two players have the same score. When the $k$ days of the competition are over, it is found that each player's total score is 26 points. Determine all possible pairs $(n, k)$.
The total score of all players is $k n(n+1) / 2-26 n$, which means $k(n+1)=52$. Therefore, $(n, k)$ can be $(3,13),(12,4),(25,2),(51,1)$. However, $(51,1)$ does not meet the requirements.

When $(n, k)=(3,13)$, the score situation for 3 players over 13 days is $(1,2,3)+2(2,3,1)+$ $2(3,1,2)+3(1,3,2)+2(3,2,1)+3(2,1,3)=(26,26,26)$. When $(n, k)=(12,4)$, the score situation for 12 players over 4 days is $2(1,2, \cdots, 11,12)+2(12,11, \cdots, 2,1)=(26$, $26, \cdots, 26)$. When $(n, k)=(25,2)$, the score situation for 25 players over 2 days is $(1,2, \cdots, 24$, 25) $+(25,24, \cdots, 2,1)-(26,26, \cdots, 26)$.