IMC 1994 · Problem 5

Let $x_1, x_2, \dots, x_k$ be vectors of $m$-dimensional Euclidian space, such that $x_1 + x_2 + \dots + x_k = 0$. Show that there exists a permutation $\pi$ of the integers $\{1, 2, \dots, k\}$ such that

$$\left\| \sum_{i=1}^{n} x_{\pi(i)} \right\| \le \left( \sum_{i=1}^{k} \|x_i\|^2 \right)^{1/2} \quad \text{for each } n = 1, 2, \dots, k.$$

Note that $\|\cdot\|$ denotes the Euclidian norm.