Question 6
For integer $n \ge 2$,
let $x_1$, $x_2$, …, $x_n$ 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 $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$.
For which choices of $x_1$, $x_2$, …, $x_n$, $\lambda$ does equality hold?
let $x_1$, $x_2$, …, $x_n$ 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 $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$.
For which choices of $x_1$, $x_2$, …, $x_n$, $\lambda$ does equality hold?