IMC 2001 · Problem 6

For each positive integer $n$, let $f_n(\vartheta) = \sin \vartheta \cdot \sin(2\vartheta) \cdot \sin(4\vartheta) \cdots \sin(2^n \vartheta)$. For all real $\vartheta$ and all $n$, prove that

$$|f_n(\vartheta)| \le \frac{2}{\sqrt{3}} |f_n(\pi/3)|.$$