Difference between revisions of "2021 IMO Problems/Problem 2"
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− | ==Problem== | + | == Problem == |
Show that the inequality | Show that the inequality | ||
<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>. | ||
− | ==Video solutions== | + | == Solution == |
+ | then, since <cmath>\sqrt{x}\geq 0, | ||
+ | \sum_{i=1}^{n}\sum_{j=1}^{n}(\sqrt{x_i-x_j}^4)\leq \sum_{i=1}^{n}\sum_{j=1}^{n}(\sqrt{x_i+x_j}^4)</cmath> | ||
+ | then, | ||
+ | <cmath>\sum \sum x_i^2+x_j^2-2x_ix_j \leq \sum \sum x_i^2+x_j^2+2x_ix_j | ||
+ | \to \sum \sum 4x_ix_j\geq 0,</cmath> | ||
+ | therefore we have to prove that | ||
+ | <cmath>\sum \sum a_ia_j\geq 0</cmath> for every list <math>x_i</math>, | ||
+ | and we can describe this to | ||
+ | <cmath>\sum \sum a_ia_j=\sum a_i^2 + \sum\sum a_ia_j(i\neq j)</cmath> | ||
+ | we know that | ||
+ | <cmath>\frac{a_i^2}{2}+\frac{a_j^2}{2} \geq |a_ia_j|</cmath> | ||
+ | therefore, <cmath>a_i^2+a_j^2 \geq -(a_ia_j+a_ja_i)</cmath> | ||
+ | <cmath>\to \sum a_i^2 + \sum\sum a_ia_j \geq 0</cmath> | ||
+ | <cmath>Q.E.D.</cmath> | ||
+ | --[[User:Mathhyhyhye|Mathhyhyhy]] 13:29, 6 June 2023 (EST) | ||
+ | |||
+ | == Video solutions == | ||
https://youtu.be/cI9p-Z4-Sc8 [Video contains solutions to all day 1 problems] | https://youtu.be/cI9p-Z4-Sc8 [Video contains solutions to all day 1 problems] | ||
+ | |||
https://youtu.be/akJOPrh5sqg [uses integral] | https://youtu.be/akJOPrh5sqg [uses integral] | ||
+ | |||
+ | https://www.youtube.com/watch?v=P9Ge8HAf6xk | ||
+ | |||
+ | == See also == | ||
+ | {{IMO box|year=2021|num-b=1|num-a=3}} |
Latest revision as of 05:11, 24 April 2024
Contents
Problem
Show that the inequality holds for all real numbers .
Solution
then, since then, therefore we have to prove that for every list , and we can describe this to we know that therefore, --Mathhyhyhy 13:29, 6 June 2023 (EST)
Video solutions
https://youtu.be/cI9p-Z4-Sc8 [Video contains solutions to all day 1 problems]
https://youtu.be/akJOPrh5sqg [uses integral]
https://www.youtube.com/watch?v=P9Ge8HAf6xk
See also
2021 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |