IMO Shortlist 1987 problem 15


Kvaliteta:
  Avg: 0.0
Težina:
  Avg: 0.0
Dodao/la: arhiva
April 2, 2012
LaTeX PDF
Let x_1,x_2,\ldots,x_n be real numbers satisfying x_1^2+x_2^2+\ldots+x_n^2=1. Prove that for every integer k\ge2 there are integers a_1,a_2,\ldots,a_n, not all zero, such that |a_i|\le k-1 for all i, and |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
Source: Međunarodna matematička olimpijada, shortlist 1987