HMMT 2008 S6

Suppose that $j(0) j(1) \cdots j(n-1)$ is a valid juggling sequence. For $i=0,1, \ldots, n-1$, Let $a_{i}$ denote the remainder of $j(i)+i$ when divided by $n$. Prove that $\left(a_{0}, a_{1}, \ldots, a_{n-1}\right)$ is a permutation of $(0,1, \ldots, n-1)$.