IMC 2004 · Problem 3

Let $D$ be the closed unit disk in the plane, and let $p_1, p_2, \dots, p_n$ be fixed points in $D$. Show that there exists a point $p$ in $D$ such that the sum of the distances of $p$ to each of $p_1, p_2, \dots, p_n$ is greater than or equal to $1$.