A Tangent Inequality Count

★ 8/10functions trigonometry Counting Modular Arithmetic Casework

Problem

Let $a \in \mathbb{R}$ , and let $m$ be the number of integer solutions in $(0, 2025)$ of the inequality

\left(\tan\frac{\pi x}{6} - a\right)\left(\tan\frac{\pi x}{6} - a - 1\right) < 0.

Determine, with proof, which of the values $0$ , $338$ , $674$ , $1012$ are impossible for $m$ .

Answer

Solution

Difficulty	8/10
Topics	functions, trigonometry, Counting, Modular Arithmetic, Casework

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