Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer, and let
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
be a positive real number such that
![x^n + y^n = 1.](/media/m/2/1/a/21a170c85ceb8ad77364a63567bffba3.png)
Prove that
Author: unknown author, Estonia
%V0
Let $n$ be a positive integer, and let $x$ and $y$ be a positive real number such that $x^n + y^n = 1.$ Prove that $$
\left(\sum^n_{k = 1} \frac {1 + x^{2k}}{1 + x^{4k}} \right) \cdot \left( \sum^n_{k = 1} \frac {1 + y^{2k}}{1 + y^{4k}} \right) < \frac{1}{(1 - x)(1 - y)} \text{.}$$
Author: unknown author, Estonia