IMO 1973 — Official problems Question 1

Let $O$ be a point on the line $l$ and $\overrightarrow{OP_1},\overrightarrow{OP_2},\ldots,\overrightarrow{OP_n}$ unit vectors such that points $P_1,P_2,\ldots,P_n$ and line $l$ lie in the same plane and all points $P_i$ lie in the same half-plane determined by $l$. Prove that if $n$ is odd, then
$$\left\|\overrightarrow{OP_1}+\overrightarrow{OP_2}+\cdots+\overrightarrow{OP_n}\right\|\ge1.$$
($\|\overrightarrow{OM}\|$ is the length of vector $\overrightarrow{OM}$).