KöMaL 2010 A503

In space, there are given some vectors $u_1$,$u_2$,...,$u_n$ and $v$ such that |$u_1$|$\ge$1, ..., |$u_n$|$\ge$1 and |$v$|$\le$1, and $u_1$+...+$u_n$=0. Show that |$u_1$-$v$|+...+|$u_n$-$v$|$\ge n$.