Question 46
Let $v_1, v_2, \dots, v_n$ be $n (\ge 2)$ unit vectors in the plane. Prove that there exist $\lambda_1, \lambda_2, \dots, \lambda_n$, each equal to $+1$ or $-1$, such that
$$
|\lambda_1 v_1 + \lambda_2 v_2 + \dots + \lambda_n v_n| \le \sqrt{2}.
$$
(Here $|v|$ denotes the length of the vector $v$.)
$$
|\lambda_1 v_1 + \lambda_2 v_2 + \dots + \lambda_n v_n| \le \sqrt{2}.
$$
(Here $|v|$ denotes the length of the vector $v$.)