OMM 2018 Problem 4

Let $n\geq 2$ be an integer. For each $k$ -tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$ , consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$ . Determine, in terms of $n$ , the maximum possible value of the product $S_1S_2\cdots S_k$ .

Proposed by Misael Pelayo