Difference between revisions of "2012 USAMO Problems/Problem 6"

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==Solution==
 
==Solution==
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==See also==
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*[[USAMO Problems and Solutions]]
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{{USAMO newbox|year=2012|num-b=5|aftertext=|after=Last Problem}}

Revision as of 18:01, 25 April 2012

Problem

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 $2^{n - 3}/\lambda^2$. For what choices of $x_1$, $x_2$, $\dots$, $x_n$, $\lambda$ does equality hold?

Solution

See also

2012 USAMO (ProblemsResources)
Preceded by
Problem 5
Last Problem
1 2 3 4 5 6
All USAMO Problems and Solutions
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