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Given real numbers $x_i$, such that $x_1 \ge x_2 \ge x_3\ge\dotsb\ge x_n$, does there exist a sequence $y_1, y_2, \dotsc ,y_n$ such that

  1. $y_1 \ge y_2 \ge y_3 \ge\dotsb\ge y_n$

  2. $x_1+\dotsb+x_i \ge y_1+\dotsb+y_i$ for $i=1,\dotsc,n-1$

  3. $x_{i+1}+\dotsb+x_n \le y_{i+1}+\dotsb+y_n$ for $i=1,\dotsb,n-1$

  4. $x_1+x_2+\dotsb+x_n \neq y_1+y_2+\dotsb+y_n$

Karamata inequality

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closed as off-topic by user258700, Shailesh, JonMark Perry, Martin R, user91500 Jul 30 '16 at 10:31

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  • $\begingroup$ $x_i$'s and $y_i$'s are real, isn't it? $\endgroup$ – InsideOut Jul 30 '16 at 10:02
  • $\begingroup$ Thank to dear @Gianluca , Yes, they are real number. $\endgroup$ – Oai Thanh Đào Jul 30 '16 at 10:16
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    $\begingroup$ @OaiThanhĐào, You should try to come up with such a sequence of $y_i$ on your own and edit your question to include what you've tried. That way your question doesn't come off as "I don't want to do this. Do it for me." And why did you post a link to the Karamata inequality? Do you think that helps? $\endgroup$ – Mike Pierce Jul 30 '16 at 13:48
  • $\begingroup$ Sorry for my english, could you help me edit? Because my english is litle @MikePierce $\endgroup$ – Oai Thanh Đào Jul 30 '16 at 13:53
  • $\begingroup$ Dear @MikePierce Thank to You very much $\endgroup$ – Oai Thanh Đào Jul 30 '16 at 13:54