ILL 1977 Problem 6

Let $x_1, x_2, \ldots , x_n \ (n \geq 1)$ be real numbers such that $0 \leq x_j \leq \pi, \ j = 1, 2,\ldots, n.$ Prove that if $\sum_{j=1}^n (\cos x_j +1)$ is an odd integer, then $\sum_{j=1}^n \sin x_j \geq 1.$