Distinct distance maximizers

Question

Let $a_1,\ldots, a_n$ be distinct points in $\mathbb{R}^n$. Do there exist distinct points $b_1,\ldots, b_n$ such that $$b_i = \text{argmax}_{b\in \{b_1,\ldots b_n\}} ||a_i - b||?$$ One argument (which doesn't work always) I had in mind was the following. Consider balls centered at the $a_i$ all with the same large enough radius such that all the balls intersect. If the boundary of each ball is part of this intersection, then you can take $b_i\in\partial B_i\cap (\cap B_i)$, and this will obviously achieve the above. But it is not necessarily the case that the boundary of each ball intersect this $\cap B_i$. Any help is appreciated!