55. (USS 2) Find all $x$ for which for all $n$,
$$
\sin x+\sin 2 x+\sin 3 x+\cdots+\sin n x \leq \frac{\sqrt{3}}{2} .
$$
55. It is enough to find all $x$ from $(0,2 \pi]$ such that the given inequality holds for all $n$.
Suppose $0\sqrt{3} / 2$.
Suppose now that $2 \pi / 3 \leq x<2 \pi$. We have
$$
\sin x+\cdots+\sin n x=\frac{\cos \frac{x}{2}-\cos \frac{2 n+1}{2} x}{2 \sin \frac{x}{2}} \leq \frac{\cos \frac{x}{2}+1}{2 \sin \frac{x}{2}}=\frac{\cot \frac{x}{4}}{2} \leq \frac{\sqrt{3}}{2} .
$$

For $x=2 \pi$ the given inequality clearly holds for all $n$. Hence, the inequality holds for all $n$ if and only if $2 \pi / 3+2 k \pi \leq x \leq 2 \pi+2 k \pi$ for some integer $k$.