RMO 2024 Problem 1

Let $n>1$ be a positive integer. Call a rearrangement $a_1,a_2, \cdots , a_n$ of $1,2, \cdots , n$ nice if for every $k = 2,3, \cdots , n$ , we have that $a_1 + a_2 + \cdots + a_k$ is not divisible by $k$ . (a) If $n>1$ is odd, prove that there is no nice arrangement of $1,2, \cdots , n$ . (b) If $n$ is even, find a nice arrangement of $1,2, \cdots , n$ .