Dot product inequality

Question

I'm looking for a pair of vectors $x \in \mathbb{R}^n$, $y \in \mathbb{R}^m$, with $x , y > 0$ (the inequality holds element-wise), such that $$ (a - \boldsymbol{1}_n)^\top x > \boldsymbol{1}_m^\top y, $$ where $\boldsymbol{1}_p$ is the $p$-dimensional vector with all the elements equal to $1$, while $a \in \mathbb{R}^n$ is some parameter. Any condition on $a$ that could guarantee the existence of such pair $x,y$? A pointer to some literature on this kind of inequalities would be much appreciated.