MathDB

More like combinatorics, i think, but it concerns vectors

Source: Brazilian Math Olympiad 2005, Problem 3

October 24, 2005

vectorlinear algebralinear algebra unsolved

Problem Statement

Let

v_1,v_2,\ldots,v_n

vectors in

\mathbb{R}^2

such that

|v_i|\leq 1

for

1 \leq i \leq n

and

\sum_{i=1}^n v_i=0

. Prove that there exists a permutation

\sigma

(1,2,\ldots,n)

such that

\left|\sum_{j=1}^k v_{\sigma(j)}\right| \leq\sqrt 5

for every

k

1\leq k \leq n

. Remark: If

v = (x,y)\in \mathbb{R}^2

|v| = \sqrt{x^2 + y^2}