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Let $1>x_1>x_2>x_3>x_4>0$ and $x_1+x_2+x_3+x_4 < 2$. Prove at least one of the following inequalities do not hold:

$$1/x_1 + 1/(x_1 - x_2) + 1/(x_1 - x_3) + 1/(x_1 - x_4) < 1/(1 - x_1)$$ $$ 1/x_2 -1/(x_1 - x_2) + 1/(x_2 - x_3) + 1/(x_2 - x_4) < 1/(1 - x_2)$$ $$1/x_3 - 1/(x_1 - x_3) - 1/(x_2 - x_3) + 1/(x_3 - x_4) < 1/(1 - x_3) $$ $$ 1/x_4 - 1/(x_1 - x_4) - 1/(x_2 - x_4) - 1/(x_3 - x_4) < 1/(1 - x_4)$$

I was able to prove the $3$ dimensional version of the problem through case work and I have some trivial bounds on potential counterexamples, namely, $x_1>4/5$, $x_2> 12/25$, $x_3 > 24/125$, $x_4 > 24/625$.

mtheorylord
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  • Computer calculations suggest that the claim is true. – Alex Ravsky May 13 '20 at 06:01
  • @mtheorylord You give $x_1 > 4/5$ etc. So you can prove the case when $x_1 < 4/5$ for example? – River Li May 19 '20 at 07:34
  • $x_1 > 1/2$ by the first equation, bounding LHS of the first equation by $4/x_1$ gives $x_1 > 4/5$. we can get the other bounds similarily. – mtheorylord May 20 '20 at 03:24
  • @mtheorylord Nice observation. – River Li May 20 '20 at 03:43
  • @mtheorylord According to your new question https://math.stackexchange.com/questions/3686293/change-along-some-direction-is-positive?noredirect=1#comment7576692_3686293, if we let $f(x) = \sum_{i=1}^4 \ln(x_i(1-x_i)) + \sum_{1\le i<j \le 4} \ln(x_i-x_j)$, then we want to prove that at least one of the partial derivatives is non-negative? – River Li May 22 '20 at 23:42
  • Yeah, that's correct. – mtheorylord May 22 '20 at 23:48

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