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Inequality with cos, tan and log

Source: Czech and Slovak Olympiad 1970, National Round, Problem 6

July 4, 2024
inequalitiestrigonometryalgebralogarithms

Problem Statement

Determine all real xx such that tan⁡(x)−1 (log⁡tan⁡(x)(2+4cos⁡2(x)−2))≥0.\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.
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