Difference between revisions of "2011 OIM Problems/Problem 5"
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| − | Let <math>x_1, \cdots , x_n</math> be positive real numbers. Prove that there exist <math>a_1, \cdots , a_n \in \left\{ | + | Let <math>x_1, \cdots , x_n</math> be positive real numbers. Prove that there exist <math>a_1, \cdots , a_n \in \left\{ -1, 1 \right\}</math> such that |
<cmath>a_1x_1^2+\cdots+a_nx_n^2\ge (a_1x_1+\cdots+a_nx_n)^2</cmath> | <cmath>a_1x_1^2+\cdots+a_nx_n^2\ge (a_1x_1+\cdots+a_nx_n)^2</cmath> | ||
Latest revision as of 15:00, 14 December 2023
Problem
Let 
be positive real numbers. Prove that there exist 
such that
![\[a_1x_1^2+\cdots+a_nx_n^2\ge (a_1x_1+\cdots+a_nx_n)^2\]](http://latex.artofproblemsolving.com/f/3/3/f33fbd157011e9f38c438d248986b67de381da06.png)
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
