IMC 2006 · Problem 4

Let $v_0$ be the zero vector in $\mathbb{R}^n$ and let $v_1, v_2, \dots, v_{n+1} \in \mathbb{R}^n$ be such that the Euclidean norm $|v_i - v_j|$ is rational for every $0 \le i, j \le n+1$. Prove that $v_1, \dots, v_{n+1}$ are linearly dependent over the rationals.