USAMO 2012 Problem 6

For integer $n \geq 2$, let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying $$x_{1}+x_{2}+\cdots+x_{n}=0 \text { and } x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$$ For each subset $A \subseteq\{1,2, \ldots, n\}$, define $S_{A}=\sum_{i \in A} x_{i}$. (If $A$ is the empty set, then $S_{A}=0$.) Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_{A} \geq \lambda$ is at most $2^{n-3} / \lambda^{2}$. For which choices of $x_{1}, x_{2}, \ldots, x_{n}, \lambda$ does equality hold?