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Ineq with n variables (IMO SL 1987-P15)

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August 19, 2010
inequalitiespigeonhole principlelinear algebrainequality systemIMO ShortlistIMO 1987IMO

Problem Statement

Let x1,x2,…,xnx_1,x_2,\ldots,x_n be real numbers satisfying x12+x22+…+xn2=1x_1^2+x_2^2+\ldots+x_n^2=1. Prove that for every integer k≥2k\ge2 there are integers a1,a2,…,ana_1,a_2,\ldots,a_n, not all zero, such that ∣ai∣≤k−1|a_i|\le k-1 for all ii, and ∣a1x1+a2x2+…+anxn∣≤(k−1)nkn−1|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}. (IMO Problem 3)
Proposed by Germany, FR
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