olymmath

ISL 2007 A3

  • Answer

Let nn be a positive integer, and let xx and yy be positive real numbers such that xn+yn=1x^{n}+y^{n}=1. Prove that (∑k=1n1+x2k1+x4k)(∑k=1n1+y2k1+y4k)<1(1−x)(1−y)\left(\sum_{k=1}^{n} \frac{1+x^{2 k}}{1+x^{4 k}}\right)\left(\sum_{k=1}^{n} \frac{1+y^{2 k}}{1+y^{4 k}}\right)<\frac{1}{(1-x)(1-y)} (Estonia)

Official text Transcription from NuminaMath olympiads, checked word for word against the official paper. Quoted with source.

Official text: IMO 2007 Shortlist booklet, page 13.

Show the answer

\left(\sum_{k=1}^{n} \frac{1+x^{2 k}}{1+x^{4 k}}\right)\left(\sum_{k=1}^{n} \frac{1+y^{2 k}}{1+y^{4 k}}\right)<\frac{1}{(1-x)(1-y)}

Solutions

Official solution Solution 1 by the IMO Problem Selection Committee

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