SIMON-MARAIS 2023 B1

Find the smallest positive real number $r$ with the following property: For every choice of 2023 unit vectors $v_1, v_2, \ldots, v_{2023} \in \mathbb{R}^2$, a point $p$ can be found in the plane such that for each subset $S$ of $\{1, 2, \ldots, 2023\}$, the sum

$$\sum_{i \in S} v_i$$

lies inside the disc $\{x \in \mathbb{R}^2 : ||x - p|| \leq r\}$.