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IMC 2006 / B4

Source: IMC 2006 day 2 problem 4

July 26, 2006
linear algebramatrixvectorIMCcollege contests

Problem Statement

Let v0v_{0} be the zero ector and let v1,...,vn+1∈Rnv_{1},...,v_{n+1}\in\mathbb{R}^{n} such that the Euclidian norm ∣vi−vj∣|v_{i}-v_{j}| is rational for all 0≤i,j≤n+10\le i,j\le n+1. Prove that v1,...,vn+1v_{1},...,v_{n+1} are linearly dependent over the rationals.
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