On zero-sum problems of new types

In this paper, we investigate zero-sum problems of new types. For example, given $2n-1$ integers $a_1,\dots ,a_{2n-1}$ not divisible by an integer $n>1$, we prove that for some nonempty $I\subseteq \{1,\dots ,2n-1\}$ with $\mathrm{\mid }I\mathrm{\mid }⩽n$, the sum ${\sum }_{i\in I}{a}_i$ is divisible by $n$ but not divisible by $n^2$. We also pose several conjectures for further research.