MathDB

IMC problem 5

Source: IMC 1994

September 24, 2014

vectorinductionIMCcollege contests

Problem Statement

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).

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