IMC 2004 · Problem 3

Let $S_n$ be the set of all sums $\sum\limits_{k=1}^{n} x_k$, where $n \ge 2$, $0 \le x_1, x_2, \dots, x_n \le \frac{\pi}{2}$ and

$$\sum_{k=1}^{n} \sin x_k = 1.$$

a) Show that $S_n$ is an interval.

b) Let $l_n$ be the length of $S_n$. Find $\lim\limits_{n \to \infty} l_n$.