Difference between revisions of "2011 OIM Problems/Problem 5"
(Created page with "== Problem == Let <math>x_1, \cdots , x_n</math> be positive real numbers. Prove that there exist <math>a1, \cdots , a_n \in \left\{−1, 1\right\}</math> such that <cmath>a_...") |
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== Problem == | == Problem == | ||
| − | Let <math>x_1, \cdots , x_n</math> be positive real numbers. Prove that there exist <math> | + | 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> | ||
Revision as of 15:58, 14 December 2023
Problem
Let 
be positive real numbers. Prove that there exist $a_1, \cdots , a_n \in \left\{−1, 1\right\}$ (Error compiling LaTeX. Unknown error_msg) 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
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