Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.
We prove it by induction on the number $2n+1$ of vectors. The base step (when we have one vector) is clear, and for the induction step we use the hypothesis for the $2n-1$ vectors obtained by disregarding the outermost two vectors. We thus get a vector with norm $\ge 1$ betwen two with norm $1$. The sum of the two vectors of norm $1$ makes an angle of $\le\frac\pi 2$ with the vector of norm $\ge 1$, so their sum has norm $\ge 1$, and we're done.
Well, you can also optimize gradually. Choose $O=(0,0)$ and let all vectors lie in the region $y\geq 0$.
Fix the first $2n$ vectors and let $w$ be their sum. We want to minimize $|v+w|$, where $|v|=1$ and $v$ has positive $y$-coordinate.
The possible locus $v+w$ is a semicircle $\Gamma$ of unit radius located above the origin. It's easy to see that the closest point of $\Gamma$ to $O$ is one of the endpoints. Thus, we by repeating the above procedure, we can assume all vectors are $(\pm 1,0)$, which trivialises the problem.