IMC 2011 · Problem 5

Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1, \dots, v_{4n-1} \in V$, there exists a sequence $1 \le i_1 < \dots < i_{2n} \le 4n-1$ of indices such that

$$v_{i_1} + \dots + v_{i_{2n}} = 0.$$