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The following inequality appeared in the AMTI contest in India, which was held a couple of days ago.

If $x$ and $y$ are positive reals such that $x^{2014}+y^{2014}=1$ prove that

$\displaystyle \left(\sum_{k=1}^{1007} \frac{1 + x^{2k}}{1+x^{4k}}\right)\left(\sum_{k=1}^{1007} \frac{1 + y^{2k}}{1+y^{4k}}\right) < \frac1{(1-x)(1-y)}.$

I have tried grouping terms and have tried A.M-G.M inequality in various ways. Nothing I do seems to work. Would someone kindly post a solution?

Thanks!

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Spai
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  • replace 2014 by, say, 2 or 6 or 10, see whether you can do those. Maybe 4. Extremely unlikely that this depends on the size of 2014, just something like it being twice odd... – Will Jagy Nov 04 '14 at 21:19
  • I have tried that.The case with 2 is trivial. Even with just two terms(case where 2014 is replaced by 4),I have no idea what to do. – Isomorphism Nov 05 '14 at 05:00
  • Did you get the solution for this problem ? – Saikat May 19 '17 at 02:49

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