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Question : Let $n$ blue points $A_i$ and $n$ red points $B_i$ ($i=1,2,...,n$) be situated on a line. Prove that $$\sum_{i,j}A_iB_j\ge \sum_{i<j}A_iA_j+\sum_{i<j}B_iB_j$$

I tried inducting on $n$ but cant proceed in the inductive step ( as usual ). Please help, if there is a direct method of approach please give hints. Is this true if all the blue points were on a different line?

  • Probably break it down into triangles and use the triangle inequality. – Gerry Myerson Nov 02 '16 at 08:11
  • Well in the original problem all points are on a line. –  Nov 02 '16 at 08:12
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    Three points on a line still make a triangle, at least as far as the triangle inequality is concerned. In fact, if you want the triangle inequality to give equality, the three points have to be on a line (and in the right order). – Arthur Nov 02 '16 at 08:54
  • Well I tried triangle inequality but don't found a way out, please help. Thanks. –  Nov 02 '16 at 12:26
  • A question is an one-dimensional version of this question, so I vote to close it as a duplicate. – Alex Ravsky Aug 01 '17 at 06:31

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