2023 Putnam A4

Let $v_1, \dots, v_{12}$ be unit vectors in $\mathbb{R}^3$ from the origin to the vertices of a regular icosahedron. Show that for every vector $v \in \mathbb{R}^3$ and every $\varepsilon > 0$, there exist integers $a_1,\dots,a_{12}$ such that $\| a_1 v_1 + \cdots + a_{12} v_{12} - v \| < \varepsilon$.

令 $v_1, \dots, v_{12}$ 为 $\mathbb{R}^3$ 中从原点到正二十面体顶点的单位向量。证明对于每个向量 $v \in \mathbb{R}^3$ 和每个 $\varepsilon > 0$，都存在整数 $a_1,\dots,a_{12}$ 使得 $\| a_1 v_1 + \cdots + a_{12} v_{12} - v \| <\varepsilon$。