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Bound on subsets

Source: 2012 USAMO problem #6

April 25, 2012
probabilityinequalitiesreal analysis2012 USAMOProbabilistic MethodHiintegration

Problem Statement

For integer n≥2n\geq2, let x1,x2,…,xnx_1, x_2, \ldots, x_n be real numbers satisfying x1+x2+…+xn=0,andx12+x22+…+xn2=1.x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.For each subset A⊆{1,2,…,n}A\subseteq\{1, 2, \ldots, n\}, defineSA=∑i∈Axi.S_A=\sum_{i\in A}x_i.(If AA is the empty set, then SA=0S_A=0.)
Prove that for any positive number λ\lambda, the number of sets AA satisfying SA≥λS_A\geq\lambda is at most 2n−3/λ22^{n-3}/\lambda^2. For which choices of x1,x2,…,xn,λx_1, x_2, \ldots, x_n, \lambda does equality hold?
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