ILL 1987 Problem 44

Let $\theta_1,\theta_2,\cdots,\theta_n$ be $n$ real numbers such that $\sin \theta_1+\sin \theta_2+\cdots+\sin \theta_n=0$ . Prove that $$|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]$$