49th Mathematical Olympiad in Ukraine

$3\pi$ is a period of the function $f(x)$ if and only if $f(x+3\pi )=f(x)$ for arbitrary real number $x$. Putting $x=0$ we obtain that $f(0)=0$, since $f(3\pi )=\cos 3\pi n\cdot \sin \frac{3\cdot 2009\pi }{n^2}$ we have $0=\cos 3\pi n\cdot \sin \frac{3\cdot 2009\pi }{n^2}\iff 0=\sin \frac{3\cdot 2009\pi }{n^2}\iff \frac{6027\pi }{n^2}=\pi k,k\in \mathbb{Z}\iff 6027:n^2$.

As $6027=3\cdot 7^2\cdot 41$ thus the possible values for $n$ can only be numbers $1$ or $7$. It is directly verified that the numbers $1$ and $7$ satisfy the same initial conditions.