MathDB

3

Part of 2014 HMIC

Problems(1)

2014 HMIC #3

Source:

12/26/2016
Fix positive integers mm and nn. Suppose that a1,a2,…,ama_1, a_2, \dots, a_m are reals, and that pairwise distinct vectors v1,…,vm∈Rnv_1, \dots, v_m\in \mathbb{R}^n satisfy ∑j≠iajvj−vi∣∣vj−vi∣∣3=0\sum_{j\neq i} a_j \frac{v_j-v_i}{||v_j-v_i||^3}=0 for i=1,2,…,mi=1,2,\dots,m. Prove that ∑1≤i<j≤maiaj∣∣vj−vi∣∣=0.\sum_{1\le i<j\le m} \frac{a_ia_j}{||v_j-v_i||}=0.