On zero-sum problems of new types

Abstract

In this paper, we investigate zero-sum problems of new types. For example, given $2n-1$ integers $a_1,\ldots,a_{2n-1}$ relatively prime to an integer $n>1$, we prove that for some nonempty $I\subseteq\{1,\ldots,2n-1\}$ the sum $\sum_{i\in I}a_i$ is divisible by $n$ but not divisible by $n^2$, and conjecture that we may further require $|I|=n$ when $n$ is even.