RMO 2024

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 $1,2, \cdots , n$ .

For a positive integer $n$ , let $R(n)$ be the sum of the remainders when $n$ is divided by $1,2, \cdots , n$ . For example, $R(4) = 0 + 0 + 1 + 0 = 1,$ $R(7) = 0 + 1 + 1 + 3 + 2 + 1 + 0 = 8$ . Find all positive integers such that $R(n) = n-1$ .