Question 1
For integer $n \ge 2$, let $x_1, x_2, \dots, x_n$ be real numbers satisfying
$$
x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1.
$$
For each subset $A \subseteq \{1, 2, \dots, n\}$, 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 $S_A \ge \lambda$ is at most $\frac{2^{n-3}}{\lambda^2}$.
For what choices of $x_1, x_2, \dots, x_n, \lambda$ does equality hold?
$$
x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1.
$$
For each subset $A \subseteq \{1, 2, \dots, n\}$, 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 $S_A \ge \lambda$ is at most $\frac{2^{n-3}}{\lambda^2}$.
For what choices of $x_1, x_2, \dots, x_n, \lambda$ does equality hold?