VMO 1980 Problem 1

Let $x_1$, $x_2$, ... , $x_n$ be real numbers in the interval [0, π] such that (1 + cos $x_1$) + (1 + cos $x_2$) + ... + (1 + cos $x_n$) is an odd integer. Show that sin $x_1$ + sin $x_2$ + ... + sin $x_n$ ≥ 1.