CMOQR 2020 Problem 2

Given a set $S$ , of integers, an optimal partition of S into sets T, U is a partition which minimizes the value $|t - u|$ , where $t$ and $u$ are the sum of the elements of $T$ and U respectively. Let $P$ be a set of distinct positive integers such that the sum of the elements of $P$ is $2k$ for a positive integer $k$ , and no subset of $P$ sums to $k$ . Either show that there exists such a $P$ with at least $2020$ different optimal partitions, or show that such a $P$ does not exist.