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IMC2015: Day 1, Problem 55. Let n≥2, let A1,A2,…,An+1 be n+1 points in the n-dimensional Euclidean space, not lying on the same hyperplane, and let B be a point strictly inside the convex hull of A1,A2,…,An+1. Prove that ∠AiBAj>90∘ holds for at least n pairs (i,j) with 1≤i<j≤n+1. Proposed by Géza Kós, Eötvös University, Budapest Hint: Use dot-product of vectors. | |||||||||
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