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Miklós Schweitzer 2000, Problem 5

Source: Miklós Schweitzer 2000

July 30, 2016
college contestsMiklos Schweitzerreal analysis

Problem Statement

Prove that for every ε>0\varepsilon >0 there exists a positive integer nn and there are positive numbers a1,…,ana_1, \ldots, a_n such that for every ε<x<2π−ε\varepsilon < x < 2\pi - \varepsilon we have ∑k=1nakcos⁡kx<−1ε∣∑k=1naksin⁡kx∣\sum_{k=1}^n a_k\cos kx < -\frac{1}{\varepsilon}\left| \sum_{k=1}^n a_k\sin kx\right|.
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