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 16:56, 25 April 2012
Problem
For integer 
, let 
, 
, 
, 
be real numbers satisfying
![\[x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1.\]](http://latex.artofproblemsolving.com/8/c/9/8c9c8a3e2c9027abba620c84789f4c9f7586f6cc.png)
For each subset 
, define
![\[S_A = \sum_{i \in A} x_i.\]](http://latex.artofproblemsolving.com/4/5/7/45717db127bc90fec3df2da18ec709f9df5a8619.png)
(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?
