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

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(If <math>A</math> is the empty set, then <math>S_A = 0</math>.) | (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? |

==Solution== | ==Solution== |

## Revision as of 17:56, 25 April 2012

## Problem

For integer , let , , , be real numbers satisfying For each subset , define (If is the empty set, then .)

Prove that for any positive number , the number of sets satisfying is at most . For what choices of , , , , does equality hold?