IMO Shortlist 2003 problem A4


Kvaliteta:
  Avg: 0,0
Težina:
  Avg: 7,0
Dodao/la: arhiva
2. travnja 2012.
LaTeX PDF
Let n be a positive integer and let x_1\le x_2\le\cdots\le x_n be real numbers.
Prove that

\left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2.
Show that the equality holds if and only if x_1, \ldots, x_n is an arithmetic sequence.
Izvor: Međunarodna matematička olimpijada, shortlist 2003