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Let $s_r(n)$ (resp. $t_r(n)$) be the least positive integer $k$ such that for any ${\\bf a}_1,\\ldots,{\\bf a}_k\\in\\mathbb Z^r$ not congruent to ${\\bf 0}=(0,\\ldots,0)$\n  modulo $n$ (resp., with all the coordinates relatively prime to $n$), there is an $I\\subseteq\\{1,\\ldots,k\\}$ with $|I|=n$ for which $\\sum_{i\\in I}{\\bf a}_i\\equiv{\\bf 0}\\pmod n$ but $\\sum_{i\\in I}{\\bf a}_i\\not\\equiv{\\bf 0}\\pmod {n^2}$. We study lower and upper bounds for $s_r(n)$ and $t_r(n)$. 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Let $s_r(n)$ (resp. $t_r(n)$) be the least positive integer $k$ such that for any ${\\bf a}_1,\\ldots,{\\bf a}_k\\in\\mathbb Z^r$ not congruent to ${\\bf 0}=(0,\\ldots,0)$\n  modulo $n$ (resp., with all the coordinates relatively prime to $n$), there is an $I\\subseteq\\{1,\\ldots,k\\}$ with $|I|=n$ for which $\\sum_{i\\in I}{\\bf a}_i\\equiv{\\bf 0}\\pmod n$ but $\\sum_{i\\in I}{\\bf a}_i\\not\\equiv{\\bf 0}\\pmod {n^2}$. We study lower and upper bounds for $s_r(n)$ and $t_r(n)$. 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