IMC 1994 Problem 5

Let $x_1, x_2,\ldots, x_k$ be vectors of $m$ -dimensional Euclidean space, such that $x_1+x_2+\ldots + x_k=0$ . Show that there exists a permutation $\pi$ of the integers $\{ 1, 2, \ldots, k \}$ such that: $$\left\lVert \sum_{i=1}^n x_{\pi (i)}\right\rVert \leq \left( \sum_{i=1}^k \lVert x_i \rVert ^2\right)^{1/2}$$ for each $n=1, 2, \ldots, k$ . Note that $\lVert \cdot \rVert$ denotes the Euclidean norm. (18 points).