1. A competition has $n \geqslant 2$ participants, lasting $k$ days. Each day, the participants' scores are $1,2, \cdots, n$, and all participants' total scores are 26 points. Find all possible pairs $(n, k)$.
1. It is easy to know that the total sum of $k$ days is
$$
k \times \frac{1}{2} n(n+1)=26 n .
$$

Therefore, $k(n+1)=52$.
Thus, $(n, k)=(51,1),(25,2),(12,4)$, $(3,13)$. Only $(51,1)$ is impossible to achieve.
For $(25,2)$, we have
$$
\begin{array}{l}
(26,26, \cdots, 26)=(1,2, \cdots, 24,25) \\
+(25,24, \cdots, 2,1) .
\end{array}
$$

For $(12,4)$, we have
$$
\begin{array}{l}
(26,26, \cdots, 26)=2(1,2, \cdots, 11,12) \\
+2(12,11, \cdots, 2,1) .
\end{array}
$$

For $(3,13)$, we have
$$
\begin{array}{l}
(26,26,26)=(1,2,3)+2(2,3,1) \\
+2(3,1,2)+3(1,3,2)+2(3,2,1) \\
+3(2,1,3)
\end{array}
$$