Difference between revisions of "2021 IMO Problems/Problem 2"
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<cmath>\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|} \le \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}</cmath> | <cmath>\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|} \le \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}</cmath> | ||
holds for all real numbers <math>x_1,x_2,\dots,x_n</math>. | holds for all real numbers <math>x_1,x_2,\dots,x_n</math>. | ||
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| + | ==Solution== | ||
==Video solutions== | ==Video solutions== | ||
Revision as of 23:04, 5 June 2023
Problem
Show that the inequality
![\[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|} \le \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]](http://latex.artofproblemsolving.com/6/2/6/6264c77345de1aea2a587da1100b58af528389ae.png)
holds for all real numbers 
.
Solution
Video solutions
https://youtu.be/cI9p-Z4-Sc8 [Video contains solutions to all day 1 problems]
https://youtu.be/akJOPrh5sqg [uses integral]
