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

Revision as of 17:56, 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?

@@ Line 7: / Line 7: @@
 (If <math>A</math> is the empty set, then <math>S_A = 0</math>.)
-Prove that for any positive number <math>\lambda</math>, the number of sets <math>A</math> satisfying <math>S_A \ge \lambda</math> is at most <math>2^{n - 3}/\lambda^2</math>.  For what choices of <math>x_1</math>, <math>x_2</math>, \dots, <math>x_n</math>, <math>\lambda</math> does equality hold?
+Prove that for any positive number <math>\lambda</math>, the number of sets <math>A</math> satisfying <math>S_A \ge \lambda</math> is at most <math>2^{n - 3}/\lambda^2</math>.  For what choices of <math>x_1</math>, <math>x_2</math>, <math>\dots</math>, <math>x_n</math>, <math>\lambda</math> does equality hold?
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Difference between revisions of "2012 USAMO Problems/Problem 6"

Revision as of 17:56, 25 April 2012

Problem

Solution